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Exponential Equations Calculator

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1

Solved example of exponential equations

$3^x=81$
2

We can take out the unknown from the exponent by applying logarithms in base $10$ to both sides of the equation

$\log \left(3^x\right)=\log \left(81\right)$
3

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$x\log \left(3\right)=\log \left(81\right)$
4

Divide both sides of the equation by $\log \left(3\right)$

$x=\frac{\log \left(81\right)}{\log \left(3\right)}$
5

Apply the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$x=\log_{3}\left(81\right)$
6

Decompose $81$ in it's prime factors

$x=\log_{3}\left(9^{2}\right)$
7

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$x=2\log_{3}\left(9\right)$
8

Decompose $9$ in it's prime factors

$x=2\log_{3}\left(3^{2}\right)$
9

Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$

$x=4\log_{3}\left(3\right)$
10

Evaluating the logarithm of base $3$ of $3$

$x=4\cdot 1$
11

Multiply $4$ times $1$

$x=4$

Final Answer

$x=4$

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