If \log_9 x + \log_3 x = 3, what is the value of x? — JAMB UTME

If $\log_9 x + \log_3 x = 3$ , what is the value of $x$ ?

The value of

x

9

Using the change of base formula,

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{1}{2}\log_3 x

. Adding

\log_3 x

yields

\frac{3}{2}\log_3 x = 3

, which simplifies to

\log_3 x = 2

. Converting to exponential form gives

x = 3^2 = 9

Express

\log_9 x

in base

3

using the change of base formula.

\log_9 x = \frac{\log_3 x}{\log_3 9} = \frac{\log_3 x}{2} = \frac{1}{2}\log_3 x

Converting all logarithmic terms to a common base (base 3) allows combining terms.

Substitute

\frac{1}{2}\log_3 x

into the given equation.

\frac{1}{2}\log_3 x + \log_3 x = 3 \implies \frac{3}{2}\log_3 x = 3

Combine like logarithmic terms.

Solve for

\log_3 x

\log_3 x = 3 \times \frac{2}{3} = 2

Multiply both sides by

\frac{2}{3}

to isolate

\log_3 x

Convert from logarithmic form to exponential form.

x = 3^2 = 9

By definition,

\log_b a = c

is equivalent to

b^c = a

Logarithms and Change of Base

Question