Problem 32 Solve. Where appropriate, includ... [FREE SOLUTION]

Chapter 12: Problem 32

Solve. Where appropriate, include approximations to three decimal places. $$ \log _{2} x=6 $$

Short Answer

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Step by step solution

Convert Log Equation to Exponential Form

Start by converting the logarithmic equation $\log_{2} x=6$ to its exponential form. Recall that $\log_b a = c$ can be rewritten as \[a = b^c\]. Therefore, the equation becomes \[x = 2^6\].

Simplify the Exponential Expression

Now simplify $2^6$. Recall that $2^6$ means multiplying 2 by itself 6 times. \[2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\].

Conclusion

From the above calculations, we find that the value of $x$ is 64.

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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 a fundamental concept in algebra that allow us to work with the exponents of numbers. When we see an equation like $\text{log}_b a = c$, it is telling us that the exponent \c\ is the power to which the base \b\ must be raised to produce the value \a\. For example, $\text{log}_2 x = 6$ means that you need to raise 2 to the power of 6 to get x.

The base b is the number being raised to a power.
The exponent is the result of the logarithmic function.
The value a is the number you get after raising the base to the exponent.

Understanding logarithmic functions is key to solving equations where the exponent is unknown but the result is known.

Exponential Form

Converting a logarithmic equation to exponential form is an important step in solving logarithmic equations. The exponential form rewrites $\text{log}_b a = c$ as \[a = b^c\]. This can often make it easier to identify and work with the unknown value.
In our example, $\text{log}_2 x = 6$ becomes $\text{x = 2^6$.
When converted, it becomes clear that we need to calculate \2^6\ to find x.
This conversion is crucial because it leverages the basic properties of exponents to simplify the problem.

Simplification

Simplification involves reducing an expression to its simplest form. For our equation $x = 2^6$, we simplify by calculating the exponential expression \[2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\].

Each step of multiplication should be done carefully to avoid mistakes.
Simplification is often the final step to find the solution clearly.
It ensures that the answer is in the most understandable form.

For our problem, once we simplified the expression, we found that \x = 64\.
This means the original logarithmic equation $\text{log}_2 x = 6$ holds true for \x = 64\.

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91影视

Solve. Where appropriate, include approximations to three decimal places. $$ \log _{2} x=6 $$

Short Answer

Step by step solution

Convert Log Equation to Exponential Form

Simplify the Exponential Expression

Conclusion

Key Concepts

Logarithmic Functions

Exponential Form

Simplification

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