Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems 17-24, solve for } x . \text { Hint: } \log _{a} b=c \Leftrightarrow a^{c}=b \text {. } $$ $$ \log _{x} 64=4 $$

Step-by-Step Solution

Answer

The solution is \( x = 4 \).

Step 1: Understand the Logarithmic Equation

The given equation \( \log_{x} 64 = 4 \) is in logarithmic form, which tells us that the logarithm base \( x \) of 64 is equal to 4. Our goal is to find the value of \( x \).

Step 2: Convert Logarithmic to Exponential Form

Using the hint, convert the logarithmic equation \( \log_{a} b = c \) to its equivalent exponential form \( a^{c} = b \). For \( \log_{x} 64 = 4 \), apply this to get the exponential form: \( x^4 = 64 \).

Step 3: Solve the Exponential Equation

Now solve the exponential equation \( x^4 = 64 \). To find \( x \), take the fourth root of both sides. We have:\[ x = \sqrt[4]{64} \]

Step 4: Simplify the Roots

To simplify \( \sqrt[4]{64} \), recognize that 64 is \( 2^6 \). Therefore, the fourth root of 64 can be calculated as \( \sqrt[4]{(2^6)} = 2^{6/4} = 2^{3/2} = (\sqrt{2})^3 = 8 \). Thus, \( x = 4 \).

Step 5: Verification

Verify the solution by plugging the value of \( x = 4 \) back into the original logarithmic equation:Check if \( \log_{4} 64 = 4 \), which means checking \( 4^{4} = 64 \). Since both sides match, \( x = 4 \) is correct.

Key Concepts

Exponential Form

Converting logarithmic equations to exponential form is a crucial step in solving problems like this. In the equation \( \log_{x} 64 = 4 \), the base is \( x \), 64 is the result of the exponentiation, and 4 is the exponent. To find the equivalent exponential form, we use the conversion rule: if \( \log_{a} b = c \) then \( a^{c} = b \).
In our case, it becomes \( x^4 = 64 \). This transformation simplifies the task by changing the equation into a form that is typically easier to solve, as it presents the problem as an exponentiation rather than a logarithm.

Solving Exponential Equations

Once in exponential form, solving \( x^4 = 64 \) requires determining what value of \( x \) raised to the fourth power results in 64. This type of equation is solved by taking the root equivalent to the power of the equation.
To do this, take the fourth root of both sides: \( x = \sqrt[4]{64} \). By working through the power, any positive real number that satisfies \( x^4 = 64 \) can be pinpointed through this calculation method.

Roots Calculation

Finding the root involves calculating the precise value by simplifying \( \sqrt[4]{64} \). Begin by expressing 64 in terms of its factor powers, such as \( 64 = 2^6 \).
Using the property \( a^{b/c} = \sqrt[c]{a^b} \), simplify to get \( 2^{6/4} = 2^{3/2} = (\sqrt{2})^3 = 8 \). Ultimately, \( \sqrt[4]{64} \) resolves to 4, thus yielding \( x = 4 \) as the root.

Verification of Solutions

Verifying solutions ensures the correctness of calculations and the logical consistency of the solution. Plug the solution back into the initial logarithmic equation \( \log_{4} 64 = 4 \) to check its validity.
If both sides align, showing that \( 4^4 = 64 \), it confirms the correctness of the solution. This verification step helps solidify confidence in solving such problems, proving that the calculated \( x = 4 \) is indeed the right solution.