If x and y are positive real numbers such that \log_x(y) = 2 and \log_4(x) + \log_2(y) = 5… — ACT

If $x$ and $y$ are positive real numbers such that $\log_x(y) = 2$ and $\log_4(x) + \log_2(y) = 5$ , what is the value of $y$ ?

Answer: 16

Applying the definition of logarithms to

\log_x(y) = 2

yields

y = x^2

. Substituting this relationship into the second equation gives

\log_4(x) + \log_2(x^2) = 5

. Using the change of base formula, we rewrite

\log_4(x)

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

, and using the power property, we rewrite

\log_2(x^2)

2\log_2(x)

. Combining the terms results in

\frac{5}{2}\log_2(x) = 5

, which simplifies to

\log_2(x) = 2

. Converting this back to exponential form gives

x = 2^2 = 4

. Substituting this value back into the relation

y = x^2

yields

y = 4^2 = 16

Apply the definition of a logarithm to the first equation.

y = x^2

By definition,

\log_b(a) = c

is equivalent to

b^c = a

Substitute

y = x^2

into the second equation.

\log_4(x) + \log_2(x^2) = 5

Substitution reduces the system to a single equation in terms of

x

Express both logarithmic terms using base 2 properties.

\frac{1}{2}\log_2(x) + 2\log_2(x) = 5

The change of base formula gives

\log_4(x) = \frac{\log_2(x)}{\log_2(4)} = \frac{1}{2}\log_2(x)

, and the power property gives

\log_2(x^2) = 2\log_2(x)

Combine the coefficients and solve for

x

\log_2(x) = 2 \Rightarrow x = 2^2 = 4

Adding the coefficients gives

\frac{5}{2}\log_2(x) = 5

. Multiplying by

\frac{2}{5}

isolates

\log_2(x)

to solve for

x

Calculate the value of

y

using the relation from Step 1.

y = 4^2 = 16

Since

y = x^2

and

x = 4

, evaluating the square gives the final answer.

Solving systems of logarithmic equations using base conversion and definition of logarithms

Estimated Time:2m 0s

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