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# Base change formula of logarithms Calculator

## Get detailed solutions to your math problems with our Base change formula of logarithms step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here!

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### Difficult Problems

1

Solved example of base change formula of logarithms

$\lim_{x\to0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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

The derivative of the constant function ($-1$) is equal to zero

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

Applying the derivative of the exponential function

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

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

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(1+x\right)}\right)$
8

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

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(x\right)}\right)$
9

The derivative of the constant function ($1$) is equal to zero

$\lim_{x\to0}\left(1.60945^x\frac{1+x}{\frac{d}{dx}\left(x\right)}\right)$
10

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

$\lim_{x\to0}\left(1.60945^x\left(1+x\right)\right)$
11

Solve the product $1.60945^x\left(1+x\right)$

$\lim_{x\to0}\left(\left(1.6094+1.6094x\right)5^x\right)$
12

Multiplying polynomials $5^x$ and $1.6094+1.6094x$

$\lim_{x\to0}\left(1.60945^x+1.6094x5^x\right)$
13

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

$\lim_{x\to0}\left(1.60945^x\right)+\lim_{x\to0}\left(1.6094x5^x\right)$
14

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

$\left(5^0\right)\left(1.6094\right)+\lim_{x\to0}\left(1.6094x5^x\right)$

Calculate the power $5^0$

$\left(1\right)\left(1.6094\right)+\lim_{x\to0}\left(1.6094x5^x\right)$

Any expression multiplied by $1$ is equal to itself

$1.6094+\lim_{x\to0}\left(1.6094x5^x\right)$
15

Simplifying

$1.6094+\lim_{x\to0}\left(1.6094x5^x\right)$
16

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

$1.6094+\left(5^0\right)\left(1.6094\right)\left(0\right)$

Any expression multiplied by $0$ is equal to $0$

$1.6094+0$
17

Simplifying

$1.6094+0$
18

Add the values $1.6094$ and $0$

$1.6094$

$1.6094$