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Base change formula of logarithms Calculator

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1

Solved example of base change formula of 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 $\displaystyle f'(x)=\frac{a'}{a}$

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

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

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

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

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

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

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

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

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

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

$\lim_{x\to0}\left(\frac{5-15x}{-3}\right)$
8

Factor by the greatest common divisor $5$

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

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

$5\lim_{x\to0}\left(\frac{1-3x}{-3}\right)$
10

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)}$

$-\frac{5}{3}\lim_{x\to0}\left(1-3x\right)$
11

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

$-\frac{5}{3}\left(\lim_{x\to0}\left(1\right)+\lim_{x\to0}\left(-3x\right)\right)$
12

Solve the product $-\frac{5}{3}\left(\lim_{x\to0}\left(1\right)+\lim_{x\to0}\left(-3x\right)\right)$

$-\frac{5}{3}\lim_{x\to0}\left(1\right)-\frac{5}{3}\lim_{x\to0}\left(-3x\right)$
13

The limit of a constant is just the constant

$-\frac{5}{3}-\frac{5}{3}\lim_{x\to0}\left(-3x\right)$
14

Factoring by $-\frac{5}{3}$

$-\frac{5}{3}\left(1+\lim_{x\to0}\left(-3x\right)\right)$
15

Evaluate the limit by replacing all occurrences of $x$ by $0$

$-\frac{5}{3}\left(1-3\cdot 0\right)$

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

$-\frac{5}{3}\left(1+0\right)$
16

Simplifying

$-\frac{5}{3}\left(1+0\right)$

Add the values $1$ and $0$

$-\frac{5}{3}\cdot 1$

Multiply $-\frac{5}{3}$ times $1$

$-\frac{5}{3}$
17

Simplifying

$-\frac{5}{3}$

Final Answer

$-\frac{5}{3}$$\,\,\left(\approx -1.6666666666666665\right)$

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