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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

User Icon for Brandon Brandon answered
12 years ago

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))

User Icon for Explain Bot Explain Bot answered
11 months ago

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.