Question11. Find the value of $x$ . $x^{\log _{2} 7}=7$

Studdy Solution

STEP 1

1. The equation $x^{\log_{2} 7} = 7$ involves an exponential expression with a logarithmic exponent.
2. We will use properties of logarithms and exponents to solve for $x$ .

STEP 2

1. Recognize the form of the equation and apply logarithmic properties.
2. Solve for $x$ .

STEP 3

Recognize that the equation $x^{\log_{2} 7} = 7$ can be approached by taking the logarithm of both sides. Use the property of logarithms that allows us to bring the exponent down as a coefficient:
Take the logarithm base 2 of both sides:
$\log_{2}(x^{\log_{2} 7}) = \log_{2}(7)$
Apply the power rule of logarithms:
$\log_{2} 7 \cdot \log_{2} x = \log_{2} 7$

STEP 4

Since $\log_{2} 7$ is a common factor on both sides, divide both sides by $\log_{2} 7$ to isolate $\log_{2} x$ :
$\log_{2} x = 1$
Convert the logarithmic equation to its exponential form to solve for $x$ :
$x = 2^1$
$x = 2$
The value of $x$ is:
$\boxed{2}$

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