Question

Suppose that $3^a = 2$ and $3^b = 5$. If \[3^x = 150,\]then write an expression for $x$ in terms of $a$ and $b$.

Answer 1

Use logarithms to solve for a.

$3^a = 2 \implies \log_3(3^a) = a\log_3(3) = \log_3(2) \implies a = \log_3(2)$

Similarly, for b and x.

$3^b = 5 \implies b = \log_3(5)$

$3^x = 150 \implies x = \log_3(150)$

Factorize 150:

150 = 2 • 3 • 5²

Then we can expand log₃(150) using the product-to-sum and exponent property,

$\log_3(150) = \log_3(2\times3\times5^2) = \log_3(2) + \log_3(3) + \log_3(5^2)$

$\implies \log_3(15) = \log_3(2) + 1 + 2 \log_3(5) \iff \boxed{x = 1 + a + 2b}$