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Base change formula of logarithms Calculator

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1

Solved example of evaluate logarithms

$\lim_{x\to0}\left(\frac{5x}{\ln\left(1-3x\right)}\right)$
2

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

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

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

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

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

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

The derivative of the linear function times a constant, is equal to the constant

$\lim_{x\to0}\left(\frac{5}{\frac{d}{dx}\left(\ln\left(1-3x\right)\right)}\right)$
4

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

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

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

$\lim_{x\to0}\left(5\frac{1-3x}{\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(-3x\right)}\right)$
6

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

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

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

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

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

$\lim_{x\to0}\left(5\frac{1-3x}{-3}\right)$
7

The derivative of the linear function times a constant, is equal to the constant

$\lim_{x\to0}\left(5\frac{1-3x}{-3}\right)$
8

Simplify the fraction

$\lim_{x\to0}\left(-\frac{5}{3}\left(1-3x\right)\right)$
9

Solve the product $-\frac{5}{3}\left(1-3x\right)$

$\lim_{x\to0}\left(-\frac{5}{3}+5x\right)$
10

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

$\lim_{x\to0}\left(-\frac{5}{3}\right)+\lim_{x\to0}\left(5x\right)$
11

The limit of a constant is just the constant

$-\frac{5}{3}+\lim_{x\to0}\left(5x\right)$
12

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

$-1.6667+5\cdot 0$

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

$-1.6667+0$
13

Simplifying

$-1.6667+0$
14

Subtract the values $0$ and $-\frac{5}{3}$

$-\frac{5}{3}$

Answer

$-\frac{5}{3}$

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