If x is a real number such that \log_4(x - 3) = \frac{1}{2} + \log_4(2), what is the value… — ACT

If $x$ is a real number such that $\log_4(x - 3) = \frac{1}{2} + \log_4(2)$ , what is the value of $x$ ?

Answer: 7

The value of

x

is 7.

By moving the logarithmic terms to the same side and applying the quotient rule, the equation simplifies to

\log_4\left(\frac{x-3}{2}\right) = \frac{1}{2}

. Converting this to exponential form yields

\frac{x-3}{2} = 4^{1/2}

. Since

4^{1/2} = 2

, we have

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

, which gives

x - 3 = 4

, or

x = 7

. Substituting

x = 7

back into the original equation shows it is a valid solution.

Group logarithmic terms on one side of the equation.

\log_4(x - 3) - \log_4(2) = \frac{1}{2}

Grouping the logarithms allows them to be combined using logarithmic properties.

Apply the quotient property of logarithms.

\log_4\left(\frac{x - 3}{2}\right) = \frac{1}{2}

The quotient property states that

\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})

Convert the equation from logarithmic form to exponential form.

\frac{x - 3}{2} = 4^{1/2}

A logarithmic equation

\log_b(y) = z

is equivalent to

b^z = y

Evaluate the fractional exponent and solve the linear equation for

x

x = 7

Since

4^{1/2} = 2

, the equation becomes

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

. Multiplying both sides by 2 gives

x - 3 = 4

, so adding 3 to both sides yields

x = 7

Solving logarithmic equations using properties of logarithms

Estimated Time:1m 30s

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