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Chapter 4: Problem 19

Write the equation in exponential form. $$ \log _{8} 64=2 $$

Short Answer

Expert verified
8^2 = 64

Step by step solution

01

- Identify the components

Identify the base, the argument, and the result in the logarithmic equation. The logarithmic equation is \(\log _{8} 64=2\). Here, the base is 8, the argument is 64, and the result is 2.
02

- Write the exponential form

To convert the logarithmic equation to exponential form, use the definition of the logarithm. The equation \(\log _{b} a = c\) is equivalent to \(b^c = a\). For this specific problem, \(b = 8\), \(a = 64\), and \(c = 2\). Therefore, write the equation as \(8^2 = 64\).

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Key Concepts

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

Understanding Logarithms
Logarithms are a way to understand and work with large numbers easily. They are the opposite of exponentiation. For example, \(\frac{1}{\frac{y}{s}}\) symbolizes the power to which a base number must be raised to get another number. In the notation \( \log_{8}64 = 2\), 8 is the base, 64 is the argument, and 2 is the result. This means that multiplying 8 by itself twice (8 squared) gives 64.

With practice, logarithms help simplify complex calculations, especially when dealing with scientific or large-scale data.
Solving Exponential Equations
Exponential equations involve expressions where variables are in the exponent. They are transformed using logs for simpler solving. For example, from the problem \( \log_{8}64 = 2\), we know that the equivalent exponential form is \(8^2 = 64\).

Understanding exponential equations allows one to switch between exponential and logarithmic forms for better manipulation and solving.
  • Identify components: base, argument, result.
  • Rewrite in exponential form using \( b^c = a\).
Algebraic Transformations
Algebraic transformations involve changing the form of equations to make them easier to solve. In our problem, one such transformation is converting a logarithmic form to exponential form.

Given \(\frac{1}{\frac{y}{s}}\), we rewrote it as \(8^2 = 64\). This simplifies understanding and computation.

Steps involved:
  • Identify base, argument, result.
  • Use logarithmic definition \(\log_{b}a = c \rightarrow b^c = a\).
Mastering these transformations is key to solving complex algebraic problems.

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Most popular questions from this chapter

Find the difference quotient \(\frac{f(x+h)-f(x)}{h} .\) Write the answers in factored form. $$f(x)=2^{x}$$

Solve for the indicated variable. \(\log E-12.2=1.44 M\) for \(E\) (used in geology)

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{5} z=3-\log _{5}(z-20)\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\ln x+\ln (x-4)=\ln (3 x-10)\)
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