Question

3ln(b) + 2ln(c)

write as a single log.

I have tried this problem so many ways. i don't know what to do with the 3 and the 2. do you multiply or use them as exponents or what

Answer 1

Hello Diana,
Remember the three basic rules when dealing with logs:

n * log(A) = log(A^n)
log(A) + log(B) = log(A*B)
log(A) - log(B) = log(A/B)

Also note that log, and ln are both logarithms and interchangable, just with base 10 for log and base of nature exponent e for ln.

Using the above the logarithm can be easily reversed into a single logarithm.

Original Problem:
3 * ln(b) + 2 * ln(c)

Move logarithmic multiplication inside the log as exponents:
ln(b^3) + ln(c^2)

Convert addition into multiplication within the logarithmic function.
ln((b^3)*(c^2))

Since b and c are not the same base you cannot simplify any further.

Therefore the solution is:
ln((b^3)*(c^2))