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
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))
To write the expression 3ln(b) + 2ln(c) as a single logarithm, you can use the properties of logarithms. Specifically, there are two properties that will be helpful in this case:
1. The product rule of logarithms: log(m) + log(n) = log(m * n)
2. The power rule of logarithms: log(m^n) = n * log(m)
Using these properties, let's rewrite the expression step by step:
Step 1: Apply the power rule to each term separately.
3ln(b) = ln(b^3) (Since 3ln(b) means ln(b) + ln(b) + ln(b) = ln(b^3))
2ln(c) = ln(c^2) (Similarly, 2ln(c) means ln(c) + ln(c) = ln(c^2))
Step 2: Apply the product rule to the two terms.
ln(b^3) + ln(c^2) = ln(b^3 * c^2)
So, the expression 3ln(b) + 2ln(c) can be written as a single logarithm ln(b^3 * c^2).
Remember, when combining logarithms, you use the product rule when adding or subtracting two logarithms, and the power rule when multiplying or dividing two logarithms.