If \log_{10} 2 = p and \log_{10} 3 = q, which of the following expressions represents \log… — JAMB UTME

If $\log_{10} 2 = p$ and $\log_{10} 3 = q$ , which of the following expressions represents $\log_{10} 18$ in terms of $p$ and $q$ ?

The expression for

\log_{10} 18

in terms of

p

and

q

p + 2q

Expanding

\log_{10} 18

\log_{10}(2 \times 3^2)

gives

\log_{10} 2 + 2\log_{10} 3

, which simplifies directly to

p + 2q

Factorize 18 into prime factors.

18 = 2 \times 3^2

Decomposing 18 into prime factors allows the application of the given logarithm values for 2 and 3.

Apply the product rule of logarithms:

\log_{10}(a \times b) = \log_{10} a + \log_{10} b

\log_{10} 18 = \log_{10}(2 \times 3^2) = \log_{10} 2 + \log_{10}(3^2)

The logarithm of a product equals the sum of the logarithms of individual factors.

Apply the power rule of logarithms:

\log_{10}(b^n) = n \log_{10} b

\log_{10}(3^2) = 2 \log_{10} 3

The logarithm of a power expression allows bringing the exponent to the front as a multiplier.

Substitute the defined values

p = \log_{10} 2

and

q = \log_{10} 3

\log_{10} 18 = p + 2q

Replaces logarithmic terms with their algebraic representations.

Logarithm Expansion Laws (Product and Power Rules)

Question