{"id":2284,"date":"2017-06-15T22:01:06","date_gmt":"2017-06-15T22:01:06","guid":{"rendered":"http:\/\/10.10.10.4:9191\/softlect\/?p=2284"},"modified":"2019-06-22T15:17:38","modified_gmt":"2019-06-22T15:17:38","slug":"logarithm-and-natural-logarithm","status":"publish","type":"post","link":"http:\/\/softlect.com\/index.php\/logarithm-and-natural-logarithm\/","title":{"rendered":"Logarithm and Natural Logarithm"},"content":{"rendered":"

The base<\/em> b logarithm<\/em> of a number is the exponent<\/em> that we need to raise the base<\/em> in order to get the number.<\/p>\n

Logarithm definition<\/p>\n

When b is raised to the power of y is equal x:<\/p>\n

b y<\/sup><\/em> = x<\/em><\/p>\n

Then the base b logarithm of x is equal to y:<\/p>\n

logb<\/sub><\/em>(x<\/em>) = y<\/em><\/p>\n

For example when:<\/p>\n

24<\/sup> = 16<\/p>\n

Then<\/p>\n

log2<\/sub>(16) = 4<\/p>\n

Logarithm as inverse function of exponential function<\/h3>\n

The logarithmic function,<\/p>\n

y <\/em>= logb<\/sub><\/em>(x<\/em>)<\/p>\n

is the inverse function of the exponential function,<\/p>\n

x <\/em>= by<\/sup><\/em><\/p>\n

So if we calculate the exponential function of the logarithm of x (x>0),<\/p>\n

f <\/em>(f <\/em>-1<\/sup>(x<\/em>)) = b<\/em>log<\/sup>b<\/em>(x<\/em>)<\/sup> = x<\/em><\/p>\n

Or if we calculate the logarithm of the exponential function of x,<\/p>\n

f <\/em>-1<\/sup>(f <\/em>(x<\/em>)) = logb<\/em><\/sub>(bx<\/sup><\/em>) = x<\/em><\/p>\n

Natural logarithm (ln)<\/h3>\n

Natural logarithm is a logarithm to the base e:<\/p>\n

ln(x<\/em>) = loge<\/sub><\/em>(x<\/em>)<\/p>\n

When e constant is the number:<\/p>\n

<\/p>\n

Inverse logarithm calculation<\/h3>\n

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:<\/p>\n

x<\/em> = log-1<\/sup>(y<\/em>) = b y<\/sup><\/em><\/p>\n

Logarithmic function<\/h3>\n

The logarithmic function has the basic form of:<\/p>\n

f <\/em>(x<\/em>) = logb<\/sub><\/em>(x<\/em>)<\/p>\n

Logarithm rules<\/p>\n

\n
\n
Rule name<\/strong><\/div>\n
Rule<\/strong><\/div>\n<\/div>\n
\n
Logarithm product rule<\/div>\n
logb<\/sub><\/em>(x \u2219 y<\/em>) = logb<\/sub><\/em>(x<\/em>) + <\/em>logb<\/sub><\/em>(y<\/em>)<\/div>\n<\/div>\n
\n
Logarithm quotient rule<\/div>\n
logb<\/sub><\/em>(x \/ y<\/em>) = logb<\/sub><\/em>(x<\/em>) – <\/em>logb<\/sub><\/em>(y<\/em>)<\/div>\n<\/div>\n
\n
Logarithm power rule<\/div>\n
logb<\/sub><\/em>(x y<\/sup><\/em>) = y \u2219 <\/em>logb<\/sub><\/em>(x<\/em>)<\/div>\n<\/div>\n
\n
Logarithm base switch rule<\/div>\n
logb<\/sub><\/em>(c<\/em>) = 1 \/ logc<\/sub><\/em>(b<\/em>)<\/div>\n<\/div>\n
\n
Logarithm base change rule<\/div>\n
logb<\/sub><\/em>(x<\/em>) = logc<\/sub><\/em>(x<\/em>) \/ logc<\/sub><\/em>(b<\/em>)<\/div>\n<\/div>\n
\n
Derivative of logarithm<\/div>\n
f <\/em>(x<\/em>) = logb<\/em><\/sub>(x<\/em>)\u21d2 f ‘ <\/em>(x<\/em>) = 1 \/ ( x<\/em> ln(b<\/em>) )<\/div>\n<\/div>\n
\n
Integral of logarithm<\/div>\n
\u222blogb<\/sub><\/em>(x<\/em>) dx<\/em> = x \u2219 <\/em>( logb<\/sub><\/em>(x<\/em>)- 1 \/ ln(b<\/em>)) + C<\/em>\n<\/div>\n<\/div>\n
\n
Logarithm of negative number<\/div>\n
logb<\/em><\/sub>(x<\/em>)is undefined when x<\/em>\u2264 0<\/div>\n<\/div>\n
\n
Logarithm of 0<\/div>\n
logb<\/em><\/sub>(0) is undefined<\/div>\n<\/div>\n
\n
Logarithm of 0<\/div>\n
<\/div>\n<\/div>\n
\n
Logarithm of 1<\/div>\n
logb<\/em><\/sub>(1) = 0<\/div>\n<\/div>\n
\n
Logarithm of the base<\/div>\n
logb<\/em><\/sub>(b<\/em>) = 1<\/div>\n<\/div>\n
\n
Logarithm of infinity<\/div>\n
lim logb<\/em><\/sub>(\u221e<\/em>) = \u221e,<\/em>when x<\/em>\u2192\u221e<\/div>\n<\/div>\n<\/div>\n

Logarithm product rule<\/p>\n

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.<\/p>\n

logb<\/sub><\/em>(x \u2219 y<\/em>) = logb<\/sub><\/em>(x<\/em>) + <\/em>logb<\/sub><\/em>(y<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(3 \u2219 <\/em>7) = log10<\/sub>(3) + <\/em>log10<\/sub>(7)<\/p>\n

Logarithm quotient rule<\/p>\n

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.<\/p>\n

logb<\/sub><\/em>(x \/ y<\/em>) = logb<\/sub><\/em>(x<\/em>) – <\/em>logb<\/sub><\/em>(y<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(3 \/ <\/em>7) = log10<\/sub>(3) – <\/em>log10<\/sub>(7)<\/p>\n

Logarithm power rule<\/p>\n

The logarithm of x raised to the power of y is y times the logarithm of x.<\/p>\n

logb<\/sub><\/em>(x y<\/sup><\/em>) = y \u2219 <\/em>logb<\/sub><\/em>(x<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(28<\/sup>) = 8\u2219 <\/em>log10<\/sub>(2)<\/p>\n

Logarithm base switch rule<\/p>\n

The base b logarithm of c is 1 divided by the base c logarithm of b.<\/p>\n

logb<\/sub><\/em>(c<\/em>) = 1 \/ logc<\/sub><\/em>(b<\/em>)<\/p>\n

For example:<\/p>\n

log2<\/sub>(8) = 1 \/ log8<\/sub>(2)<\/p>\n

Logarithm base change rule<\/p>\n

The base b logarithm of x is base c logarithm of x divided by the base c logarithm of b.<\/p>\n

logb<\/sub><\/em>(x<\/em>) = logc<\/sub><\/em>(x<\/em>) \/ logc<\/sub><\/em>(b<\/em>)<\/p>\n

For example, in order to calculate log2<\/sub>(8) in calculator, we need to change the base to 10:<\/p>\n

log2<\/sub>(8) = log10<\/sub>(8) \/ log10<\/sub>(2)<\/p>\n

Logarithm of negative number<\/p>\n

The base b real logarithm of x when x<=0 is undefined when x is negative or equal to zero:<\/p>\n

logb<\/em><\/sub>(x<\/em>) is undefined when x<\/em> \u2264 0<\/p>\n

Logarithm of 0<\/p>\n

The base b logarithm of zero is undefined:<\/p>\n

logb<\/em><\/sub>(0) is undefined<\/p>\n

The limit of the base b logarithm of x, when x approaches zero, is minus infinity:<\/p>\n

\"\\lim_{x\\to<\/p>\n

Logarithm of 1<\/p>\n

The base b logarithm of one is zero:<\/p>\n

logb<\/em><\/sub>(1) = 0<\/p>\n

For example, teh base two logarithm of one is zero:<\/p>\n

log2<\/sub>(1) = 0<\/p>\n

Logarithm of infinity<\/p>\n

The limit of the base b logarithm of x, when x approaches infinity, is equal to infinity:<\/p>\n

lim logb<\/em><\/sub>(x<\/em>) = \u221e, when x<\/em>\u2192\u221e<\/p>\n

Logarithm of the base<\/p>\n

The base b logarithm of b is one:<\/p>\n

logb<\/em><\/sub>(b<\/em>) = 1<\/p>\n

For example, the base two logarithm of two is one:<\/p>\n

log2<\/sub>(2) = 1<\/p>\n

Logarithm derivative<\/p>\n

When<\/p>\n

f <\/em>(x<\/em>) = logb<\/sub><\/em>(x<\/em>)<\/p>\n

Then the derivative of f(x):<\/p>\n

f ‘ <\/em>(x<\/em>) = 1 \/ ( x<\/em> ln(b<\/em>) )<\/p>\n

Logarithm integral<\/p>\n

The integral of logarithm of x:<\/p>\n

\u222blogb<\/sub><\/em>(x<\/em>) dx<\/em> = x \u2219 <\/em>( logb<\/sub><\/em>(x<\/em>)- 1 \/ ln(b<\/em>)) + C<\/em><\/p>\n

For example:<\/p>\n

\u222blog2<\/sub>(x<\/em>) dx<\/em> = x \u2219 <\/em>( log2<\/sub>(x<\/em>)- 1 \/ ln(2)) + C<\/em><\/p>\n

Natural logarithm is the logarithm to the base e of a number.<\/p>\n

Definition of natural logarithm<\/p>\n

When<\/p>\n

e y<\/sup><\/em> = x<\/em><\/p>\n

Then base e logarithm of x is<\/p>\n

ln(x<\/em>) = loge<\/em><\/sub>(x<\/em>) = y<\/em><\/p>\n

The e constant or Euler’s number is:<\/p>\n

e<\/em> \u2248 2.71828183<\/p>\n

Ln as inverse function of exponential function<\/h3>\n

The natural logarithm function ln(x) is the inverse function of the exponential function ex<\/sup>.<\/p>\n

For x>0,<\/p>\n

f <\/em>(f <\/em>-1<\/sup>(x<\/em>)) = e<\/em>ln(x<\/em>)<\/sup> = x<\/em><\/p>\n

Or<\/p>\n

f <\/em>-1<\/sup>(f <\/em>(x<\/em>)) = ln(ex<\/sup><\/em>) = x<\/em><\/p>\n

Natural logarithm rules and properties<\/p>\n

\n
\n
Rule name<\/strong><\/div>\n
Rule<\/strong><\/div>\n
Example<\/strong><\/div>\n<\/div>\n
\n
Product rule<\/div>\n
ln(x \u2219 y<\/em>) = ln(x<\/em>) + <\/em>ln(y<\/em>)<\/div>\n
ln(3 \u2219 <\/em>7) = ln(3) + <\/em>ln(7)<\/div>\n<\/div>\n
\n
Quotient rule<\/div>\n
ln(x \/ y<\/em>) = ln(x<\/em>) – <\/em>ln(y<\/em>)<\/div>\n
ln(3 \/ <\/em>7) = ln(3) – <\/em>ln(7)<\/div>\n<\/div>\n
\n
Power rule<\/div>\n
ln(x y<\/sup><\/em>) = y \u2219 <\/em>ln(x<\/em>)<\/div>\n
ln(28<\/sup>) = 8\u2219 <\/em>ln(2)<\/div>\n<\/div>\n
\n
ln derivative<\/div>\n
f <\/em>(x<\/em>) = ln(x<\/em>)\u21d2 f ‘ <\/em>(x<\/em>) = 1 \/ x<\/em><\/p>\n<\/div>\n
<\/div>\n<\/div>\n
\n
ln integral<\/div>\n
\u222bln(x<\/em>)dx<\/em> = x \u2219 <\/em>(ln(x<\/em>) – 1) + C<\/em><\/div>\n
<\/div>\n<\/div>\n
\n
ln of negative number<\/div>\n
ln(x<\/em>) is undefined when x <\/em>\u2264 0<\/div>\n
<\/div>\n<\/div>\n
\n
ln of zero<\/div>\n
ln(0) is undefined<\/div>\n
<\/div>\n<\/div>\n
\n
ln of zero<\/div>\n
\"\"<\/div>\n
<\/div>\n<\/div>\n
\n
ln of one<\/div>\n
ln(1) = 0<\/div>\n
<\/div>\n<\/div>\n
\n
ln of infinity<\/div>\n
lim ln(x<\/em>) = \u221e ,when x<\/em>\u2192\u221e<\/div>\n
<\/div>\n<\/div>\n<\/div>\n

Logarithm product rule<\/h4>\n

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.<\/p>\n

logb<\/sub><\/em>(x \u2219 y<\/em>) = logb<\/sub><\/em>(x<\/em>) + <\/em>logb<\/sub><\/em>(y<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(3 \u2219 <\/em>7) = log10<\/sub>(3) + <\/em>log10<\/sub>(7)<\/p>\n

Logarithm quotient rule<\/p>\n

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.<\/p>\n

logb<\/sub><\/em>(x \/ y<\/em>) = logb<\/sub><\/em>(x<\/em>) – <\/em>logb<\/sub><\/em>(y<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(3 \/ <\/em>7) = log10<\/sub>(3) – <\/em>log10<\/sub>(7)<\/p>\n

Logarithm power rule<\/p>\n

The logarithm of x raised to the power of y is y times the logarithm of x.<\/p>\n

logb<\/sub><\/em>(x y<\/sup><\/em>) = y \u2219 <\/em>logb<\/sub><\/em>(x<\/em>)<\/p>\n

For example:<\/p>\n

log10<\/sub>(28<\/sup>) = 8\u2219 <\/em>log10<\/sub>(2)<\/p>\n

Derivative of natural logarithm<\/p>\n

The derivative of the natural logarithm function is the reciprocal function.<\/p>\n

When<\/p>\n

f <\/em>(x<\/em>) = ln(x<\/em>)<\/p>\n

The derivative of f(x) is:<\/p>\n

f ‘ <\/em>(x<\/em>) = 1 \/ x<\/em><\/p>\n

Integral of natural logarithm<\/p>\n

The integral of the natural logarithm function is given by:<\/p>\n

When<\/p>\n

f <\/em>(x<\/em>) = ln(x<\/em>)<\/p>\n

The integral of f(x) is:<\/p>\n

\u222b f <\/em>(x<\/em>)dx<\/em> = \u222bln(x<\/em>)dx<\/em> = x \u2219 <\/em>(ln(x<\/em>) – 1) + C<\/em><\/p>\n

Ln of 0<\/p>\n

The natural logarithm of zero is undefined:<\/p>\n

ln(0) is undefined<\/p>\n

The limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:<\/p>\n

\"\"<\/p>\n

Ln of 1<\/p>\n

The natural logarithm of one is zero:<\/p>\n

ln(1) = 0<\/p>\n

Ln of infinity<\/p>\n

The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:<\/p>\n

lim ln(x<\/em>) = \u221e, when x<\/em>\u2192\u221e<\/p>\n","protected":false},"excerpt":{"rendered":"

The base b logarithm of a number is the exponent that we need to raise the base in order to get the number. Logarithm definition When b is raised to… <\/p>\n","protected":false},"author":1,"featured_media":3120,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[34],"tags":[],"aioseo_notices":[],"amp_enabled":true,"_links":{"self":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2284"}],"collection":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/comments?post=2284"}],"version-history":[{"count":12,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2284\/revisions"}],"predecessor-version":[{"id":3556,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/posts\/2284\/revisions\/3556"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/media\/3120"}],"wp:attachment":[{"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/media?parent=2284"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/categories?post=2284"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/softlect.com\/index.php\/wp-json\/wp\/v2\/tags?post=2284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}