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Evaluate logarithms Calculator

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1

Solved example of Logarithmic differentiation

$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$
2

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x-1\right)}{\frac{d}{dx}\left(\ln\left(1+x\right)\right)}\right)$
3

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x-1\right)}{\frac{1}{1+x}\cdot\frac{d}{dx}\left(1+x\right)}\right)$
4

The derivative of a sum of two functions is the sum of the derivatives of each function

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x\right)+\frac{d}{dx}\left(-1\right)}{\frac{1}{1+x}\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(x\right)\right)}\right)$
5

The derivative of the constant function is equal to zero

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x\right)+0}{\frac{1}{1+x}\cdot\frac{d}{dx}\left(x\right)}\right)$
6

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(5^x\right)}{\frac{1}{1+x}}\right)$
7

Applying the derivative of the exponential function

$\lim_{x\to0}\left(\frac{\sqrt[4]{45}5^x\frac{d}{dx}\left(x\right)}{\frac{1}{1+x}}\right)$
8

The derivative of the linear function is equal to $1$

$\lim_{x\to0}\left(\frac{\sqrt[4]{45}5^x}{\frac{1}{1+x}}\right)$
9

Simplifying the fraction

$\lim_{x\to0}\left(\sqrt[4]{45}5^x\left(1+x\right)\right)$
10

Solve the product $\sqrt[4]{45}5^x\left(1+x\right)$

$\lim_{x\to0}\left(5^x\left(\sqrt[4]{45}+\sqrt[4]{45}x\right)\right)$
11

Multiplying polynomials $5^x$ and $\sqrt[4]{45}+\sqrt[4]{45}x$

$\lim_{x\to0}\left(\sqrt[4]{45}5^x+\sqrt[4]{45}5^x\cdot x\right)$
12

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\lim_{x\to0}\left(\sqrt[4]{45}5^x\right)+\lim_{x\to0}\left(\sqrt[4]{45}5^x\cdot x\right)$
13

Evaluating the limit when $x$ tends to $0$

$5^0\sqrt[4]{45}+\lim_{x\to0}\left(\sqrt[4]{45}5^x\cdot x\right)$
14

Simplifying

$\sqrt[4]{45}+\lim_{x\to0}\left(\sqrt[4]{45}5^x\cdot x\right)$
15

Evaluating the limit when $x$ tends to $0$

$\sqrt[4]{45}+5^0\sqrt[4]{45}0$
16

Simplifying

$\sqrt[4]{45}+0$
17

Add the values $\sqrt[4]{45}$ and $0$

$\sqrt[4]{45}$

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