Solve (4/5) ^ 6 * (4/5) ^ - 14 = (4/5) ^ (2x) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(\frac{4}{5})^{6}\times(\frac{4}{5})^{-14}=(\frac{4}{5})^{2x}$$

Solve for x

$x=-4$

Steps Using Definition of Logarithm

To multiply powers of the same base, add their exponents. Add $6$ and $-14$ to get $-8$.

$$\left(\frac{4}{5}\right)^{-8}=\left(\frac{4}{5}\right)^{2x}$$

Calculate $\frac{4}{5}$ to the power of $-8$ and get $\frac{390625}{65536}$.

$$\frac{390625}{65536}=\left(\frac{4}{5}\right)^{2x}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\frac{4}{5}\right)^{2x}=\frac{390625}{65536}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{4}{5}\right)^{2x})=\log(\frac{390625}{65536})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$2x\log(\frac{4}{5})=\log(\frac{390625}{65536})$$

Divide both sides by $\log(\frac{4}{5})$.

$$2x=\frac{\log(\frac{390625}{65536})}{\log(\frac{4}{5})}$$

By the change-of-base formula $\frac{\log(a)}{\log(b)}=\log_{b}\left(a\right)$.

$$2x=\log_{\frac{4}{5}}\left(\frac{390625}{65536}\right)$$

Divide both sides by $2$.

$$x=-\frac{8}{2}$$

Solve for x (complex solution)

$x=\frac{\pi n_{1}i}{\ln(\frac{4}{5})}-\log_{\frac{4}{5}}\left(\frac{256}{625}\right)$
$n_{1}\in \mathrm{Z}$