Solve (3/4) ^ 6 * (16/5) ^ 5 = (4/3) ^ (x + 2) | 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

$$(\frac{3}{4})^{6}\times(\frac{16}{5})^{5}=(\frac{4}{3})^{x+2}$$

Solve for x

$x=-\log_{\frac{4}{3}}\left(\frac{3125}{104976}\right)\approx 12.215907093$

Steps Using Definition of Logarithm

Calculate $\frac{3}{4}$ to the power of $6$ and get $\frac{729}{4096}$.

$$\frac{729}{4096}\times \left(\frac{16}{5}\right)^{5}=\left(\frac{4}{3}\right)^{x+2}$$

Calculate $\frac{16}{5}$ to the power of $5$ and get $\frac{1048576}{3125}$.

$$\frac{729}{4096}\times \frac{1048576}{3125}=\left(\frac{4}{3}\right)^{x+2}$$

Multiply $\frac{729}{4096}$ and $\frac{1048576}{3125}$ to get $\frac{186624}{3125}$.

$$\frac{186624}{3125}=\left(\frac{4}{3}\right)^{x+2}$$

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

$$\left(\frac{4}{3}\right)^{x+2}=\frac{186624}{3125}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{4}{3}\right)^{x+2})=\log(\frac{186624}{3125})$$

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

$$\left(x+2\right)\log(\frac{4}{3})=\log(\frac{186624}{3125})$$

Divide both sides by $\log(\frac{4}{3})$.

$$x+2=\frac{\log(\frac{186624}{3125})}{\log(\frac{4}{3})}$$

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

$$x+2=\log_{\frac{4}{3}}\left(\frac{186624}{3125}\right)$$

Subtract $2$ from both sides of the equation.

$$x=\frac{\ln(\frac{186624}{3125})}{\ln(\frac{4}{3})}-2$$

Solve for x (complex solution)

$x=\frac{2\pi n_{1}i}{\ln(\frac{4}{3})}-\log_{\frac{4}{3}}\left(\frac{3125}{104976}\right)$
$n_{1}\in \mathrm{Z}$