Problem 66 Solve equation. \(\log _{x} 64... [FREE SOLUTION]

Chapter 10: Problem 66

Solve equation. \(\log _{x} 64=2\)

Short Answer

Expert verified

x = 8

Step by step solution

Convert Logarithmic to Exponential Form

The given equation is \(\backslash log _{x} 64=2\). To solve for x, convert the logarithmic equation to its exponential form: \[x^2 = 64 . \]

Solve the Exponential Equation

Solve the exponential equation \x^2 = 64 by taking the square root of both sides. This gives: \[x = \sqrt{64} \text{or } x = -\sqrt{64} \] Simplify to find: \[x = 8 \text{or } x = - 8 \]

Determine the Valid Solution

Because the base of a logarithm must be a positive number greater than 1, discard -8. Therefore, the valid solution is: \[ x = 8 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. This means they help us solve equations where the exponent is unknown. If we have an equation like \(\backslash log _{x} 64=2\), it asks 'to what power do we need to raise x to get 64?' The answer to this question is converted back into exponential form for easier solving.

The base of a logarithm must be greater than 0.
The input to the log function must be greater than 0.
In our example, the base x is what we need to determine.

Exponential Equations

An exponential equation is one where the variable appears in the exponent. When dealing with logarithms, we often convert the log equation to an exponential form. This makes it easier to solve.
For instance, with \(\backslash log _{x} 64=2 \), we convert it to: \[ x^2 = 64. \]

From this, we solve the simple exponential equation by taking the square root of both sides.
This results in \({x = \backslash pm \backslash sqrt{64}} \).
That simplifies to \({x = 8 \text{ or } x = -8} \).

Valid Solutions in Logarithms

Determining the valid solution for logarithmic equations involves some additional checks. It's essential that the base of the logarithm, which is x in our case, be a positive number greater than 1. This constraint arises because:

Bases of logarithms cannot be negative or 1, as the log function would not be defined properly.

When we solved \(x^2 = 64\), we got \(x = 8 \text{ or } x = -8\).

Since the base must be positive and greater than 1, \(-8\) is not valid.

Therefore, the only valid solution is \(x = 8\). Always remember to check back with the logarithm's base restriction rules to confirm solutions.

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91影视

Solve equation. \(\log _{x} 64=2\)

Short Answer

Step by step solution

Convert Logarithmic to Exponential Form

Solve the Exponential Equation

Determine the Valid Solution

Key Concepts

Logarithmic Functions

Exponential Equations

Valid Solutions in Logarithms

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