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As it is, this logarithm cannot be found. No calculator "knows" base 8 logarithms and 32 is not a known power of 8. So what we need to do is transform this logarithm into an expression we can find. Fortunately there is a base conversion formula, , which we can use to change this base 8 logarithm into a logarithm we can find.
So we need to choose a base to change the base 8 logarithm into. I'll choose two different bases.
Base 2. Since 8 is a power of 2 (), using base 2 has some promise of making the problem easier. Using the conversion formula to convert the base 8 logarithm into an expression of base 2 logarithms we get:
As stated earlier 8 is 2 cubed. So the logarithm in the denominator is 3. And, as it turns out 32 is also a power of 2! . So the logarithm in the numerator is 5. This makes our expression:
5/3
And we are finished (except perhaps changing the improper fraction into a mixed number).
If 8 had not been a power of 2 or if we didn't recognize that 8 (or 32) was a power of 2, then we would need to cahnge the base to one our calculator "knows" (like base 10 or base e (ln)). Converting your expression into base 10 logarithms we get:
And now we would use our calculators to find the two logarithms and them divide. The first way is better than this way. For one, calculators use decimal approximations for most logarithms. Secondly, the decimal answer you get from a calculator may by hard to recognize as equal to the fraction 5/3.
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