Solve (33/17) ^ 15 + (33/17) ^ 8 = (33/77) ^ (4x - 9) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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Question

$$(\frac{33}{17})^{15}+(\frac{33}{17})^{8}=(\frac{33}{77})^{4x-9}$$

Solve for x

$x=-\frac{\log_{\frac{3}{7}}\left(\frac{115509094888367863153115351}{1191137399257048369396384950}\right)}{4}\approx -0.688457615$

Steps Using Definition of Logarithm

Calculate $\frac{33}{17}$ to the power of $15$ and get $\frac{59938945498865420543457}{2862423051509815793}$.

$$\frac{59938945498865420543457}{2862423051509815793}+\left(\frac{33}{17}\right)^{8}=\left(\frac{33}{77}\right)^{4x-9}$$

Calculate $\frac{33}{17}$ to the power of $8$ and get $\frac{1406408618241}{6975757441}$.

$$\frac{59938945498865420543457}{2862423051509815793}+\frac{1406408618241}{6975757441}=\left(\frac{33}{77}\right)^{4x-9}$$

Add $\frac{59938945498865420543457}{2862423051509815793}$ and $\frac{1406408618241}{6975757441}$ to get $\frac{60516049344970196077650}{2862423051509815793}$.

$$\frac{60516049344970196077650}{2862423051509815793}=\left(\frac{33}{77}\right)^{4x-9}$$

Reduce the fraction $\frac{33}{77}$ to lowest terms by extracting and canceling out $11$.

$$\frac{60516049344970196077650}{2862423051509815793}=\left(\frac{3}{7}\right)^{4x-9}$$

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

$$\left(\frac{3}{7}\right)^{4x-9}=\frac{60516049344970196077650}{2862423051509815793}$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{3}{7}\right)^{4x-9})=\log(\frac{60516049344970196077650}{2862423051509815793})$$

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

$$\left(4x-9\right)\log(\frac{3}{7})=\log(\frac{60516049344970196077650}{2862423051509815793})$$

Divide both sides by $\log(\frac{3}{7})$.

$$4x-9=\frac{\log(\frac{60516049344970196077650}{2862423051509815793})}{\log(\frac{3}{7})}$$

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

$$4x-9=\log_{\frac{3}{7}}\left(\frac{60516049344970196077650}{2862423051509815793}\right)$$

Add $9$ to both sides of the equation.

$$4x=\frac{\ln(\frac{60516049344970196077650}{2862423051509815793})}{\ln(\frac{3}{7})}-\left(-9\right)$$

Divide both sides by $4$.

$$x=\frac{\frac{\ln(\frac{60516049344970196077650}{2862423051509815793})}{\ln(\frac{3}{7})}+9}{4}$$

Solve for x (complex solution)

$x=\frac{\pi n_{1}i}{2\ln(\frac{3}{7})}-\frac{\log_{\frac{3}{7}}\left(\frac{115509094888367863153115351}{1191137399257048369396384950}\right)}{4}$
$n_{1}\in \mathrm{Z}$