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# Base change formula of logarithms Calculator

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###  Difficult Problems

1

Solved example of base change formula of logarithms

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

Plug in the value $0$ into the limit

$\frac{5^0-1}{\ln\left(1+0\right)}$

Calculate the power $5^0$

$\frac{1-1}{\ln\left(1+0\right)}$

Subtract the values $1$ and $-1$

$\frac{0}{\ln\left(1+0\right)}$

Add the values $1$ and $0$

$\frac{0}{\ln\left(1\right)}$

Calculating the natural logarithm of $1$

$\frac{0}{0}$
2

If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
3

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 0}\left(\frac{\frac{d}{dx}\left(5^x-1\right)}{\frac{d}{dx}\left(\ln\left(1+x\right)\right)}\right)$

Find the derivative of the numerator

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

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

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

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

$\frac{d}{dx}\left(5^x\right)$

Applying the derivative of the exponential function

$\ln\left(5\right)5^x\frac{d}{dx}\left(x\right)$

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

$\ln\left(5\right)5^x$

Calculating the natural logarithm of $5$

$\ln\left(5\right)5^x$

Find the derivative of the denominator

$\frac{d}{dx}\left(\ln\left(1+x\right)\right)$

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{1+x}\frac{d}{dx}\left(1+x\right)$

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

$\frac{1}{1+x}\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(x\right)\right)$

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

$\frac{1}{1+x}\frac{d}{dx}\left(x\right)$

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

$\frac{1}{1+x}$

Divide fractions $\frac{\ln\left(5\right)5^x}{\frac{1}{1+x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$

$\lim_{x\to0}\left(\ln\left(5\right)5^x\left(1+x\right)\right)$
4

After deriving both the numerator and denominator, the limit results in

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

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

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

Evaluate the limit $\lim_{x\to0}\left(5^x\left(1+x\right)\right)$ by replacing all occurrences of $x$ by $0$

$\ln\left(5\right)\cdot 5^0\cdot \left(1+0\right)$
7

Add the values $1$ and $0$

$\ln\left(5\right)\cdot 5^0$
8

Calculate the power $5^0$

$\ln\left(5\right)$

$\ln\left(5\right)$