If log 3 = A and log 7 = B, find log7 (9) in terms of A and B.

Verified step by step guidance

1. Recall the change of base formula for logarithms: $\log a_b=\frac{\log }{c}\frac{\log }{c}$, which means $\log a_b=\frac{\log }{c}\frac{\log }{c}$ for any positive base $c$ not equal to 1.
2. Apply the change of base formula to $\log 7_9$, choosing base 10 (common logarithm) for convenience: $\log 7_9=\frac{\log }{\mathrm{10}}\frac{\log }{\mathrm{10}}$.
3. Express $\log 9$ in terms of $\log 3$ using the property $\log \mathrm{a^n}=\; n\log \mathrm{a}$. Since $9\; =\; 3^2$, we have $\log 9=\; 2\log 3$.
4. Substitute the given values: $\log 3=\; A$ and $\log 7=\; B$ into the expression from step 2 and 3, so that $\log 7_9=\frac{}{}B$.
5. Write the final expression for $\log 7_9$ in terms of A and B as $\log 7_9=\frac{}{}B$.

Key Concepts

Change of Base Formula

The change of base formula allows you to rewrite logarithms with any base in terms of logarithms with a different base, typically base 10 or e. It states that log_b(x) = log_c(x) / log_c(b), which is essential for expressing log base 7 of 9 in terms of log base 10 values A and B.

Properties of Logarithms

Logarithmic properties such as log(xy) = log x + log y and log(x^n) = n log x help simplify expressions. Recognizing that 9 = 3^2 allows rewriting log_7(9) as log_7(3^2) = 2 log_7(3), facilitating the use of given values A and B.

Expressing Logarithms in Terms of Given Variables

Given log 3 = A and log 7 = B, these represent logarithms with a common base (usually 10). Using the change of base formula, log_7(3) can be expressed as log(3)/log(7) = A/B, enabling the expression of log_7(9) in terms of A and B.