To solve the exponential equation (5^{x+8} = 4), we need to use logarithms.
Taking the logarithm of both sides, you can use any base, but common choices are natural logarithm (base (e)) or logarithm base 10. Let’s use natural logarithm ((ln)):
[ ln(5^{x+8}) = ln(4) ]
Using the logarithm power rule ((ln(a^b) = bln(a))), the equation simplifies to:
[ (x+8)ln(5) = ln(4) ]
Now, divide both sides by (ln(5)) to solve for (x):
[ x + 8 = frac{ln(4)}{ln(5)} ]
Finally, subtract 8 from both sides to get the value of (x):
[ x = frac{ln(4)}{ln(5)} – 8 ]
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