Some calculators cannot use an arbitrary base for a logarithm, what can you do in those cases?
See solution.
Practice makes perfect
We want to find two ways to calculate log_7 12. Some calculators cannot use an arbitrary base to evaluate a logarithmic expression, but in cases like that we can use the change of base formula.
If b and c are positive real numbers with b≠1 and c ≠1, then the relation shown below holds.
log_c a = log_b a/log_b c
In particular, log_c a = log alog c and log_c a = ln aln c.
This formula allow us to rewrite a logarithm of an arbitrary base as a quotient of logarithms of any base — in particular, in terms of common logarithms log or natural logarithms ln, which are more common in calculators. Let's rewrite our expression in terms of common logarithms first.
log_7 12 = log 12/log 7
Now, in terms of natural logarithms.
log_7 12 = ln 12/ln 7
Notice that we could do the same using any different base, or we could rewrite it as an equivalent expression using the Properties of Logarithms. However, it wouldn't be particularly useful for this case, nor would it make it easier to evaluate the expression using any calculator.
