If x is a positive real number such that \log_2(x) + \log_4(x) + \log_{16}(x) = 7, what is… — ACT

If $x$ is a positive real number such that $\log_2(x) + \log_4(x) + \log_{16}(x) = 7$ , what is the value of $x$ ?

Answer: 16

The value of

x

16

By converting all logarithms to base 2, we write

\log_4(x)

\frac{1}{2}\log_2(x)

and

\log_{16}(x)

\frac{1}{4}\log_2(x)

. Summing these gives

\frac{7}{4}\log_2(x) = 7

, which simplifies to

\log_2(x) = 4

. Converting to exponential form, we find

x = 2^4 = 16

Apply the change of base formula to express the logarithms with a common base of 2.

\log_4(x) = \frac{1}{2}\log_2(x)

and

\log_{16}(x) = \frac{1}{4}\log_2(x)

, giving the equation

\log_2(x) + \frac{1}{2}\log_2(x) + \frac{1}{4}\log_2(x) = 7

Rewriting the terms with a common base allows them to be combined algebraically.

Combine the coefficients of the like terms on the left-hand side.

\frac{7}{4}\log_2(x) = 7

The sum of the coefficients is

1 + \frac{1}{2} + \frac{1}{4} = \frac{4}{4} + \frac{2}{4} + \frac{1}{4} = \frac{7}{4}

Isolate the logarithm term by dividing or multiplying by the reciprocal coefficient.

\log_2(x) = 4

Multiplying both sides by

\frac{4}{7}

solves for the value of

\log_2(x)

Convert the equation from logarithmic form to its equivalent exponential form.

x = 2^4 = 16

By definition,

\log_b(a) = c

is equivalent to

b^c = a

Change of Base Formula for Logarithms

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