# Solve the following exponential equation by expressing each side as a power of t

Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
$$\displaystyle{6}^{{{x}-{6}}}={\frac{{{1}}}{{{7776}}}}$$

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Brighton
We know that
$$\displaystyle{6}^{{8}}={7776}\Rightarrow{6}^{{-{5}}}={\frac{{{1}}}{{{7776}}}}$$
If bases are same then power must be equal.
Given equation is
$$\displaystyle{6}^{{{x}-{6}}}={\frac{{{1}}}{{{7776}}}}$$
Using equation we can write as
$$\displaystyle{6}^{{{x}-{6}}}={\frac{{{1}}}{{{7776}}}}\Rightarrow{6}^{{{x}-{6}}}={6}^{{-{5}}}$$
As bases are same powers must be equal so
$$\displaystyle\Rightarrow{x}-{6}=-{5}$$
$$\displaystyle\Rightarrow{x}={1}$$
Hence the equation is solved and the value of x is obtained as 1.