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Chapter 4: Problem 64

Simplify the expression. $$ 4^{\log _{4}(a-c)} $$

Short Answer

Expert verified
a - c

Step by step solution

01

Understand the log property

Recall the logarithmic identity that states: If you have an expression of the form \(b^{\text{log}_b(x)}\), it simplifies to just \(x\). The base \(b\) and the base of the logarithm should be the same.
02

Apply the property to the given expression

The given expression is \(4^{\text{log}_4(a-c)}\). Here, the base of the exponent is 4, and the base of the logarithm is also 4. According to the identity from Step 1, this expression simplifies directly to \(a - c\).

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Key Concepts

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

Logarithmic Identity
The logarithmic identity is a powerful tool in algebra. It states that any expression of the form \[ b^{\log_b(x)} \] simplifies directly to \[ x \]. Here, the base of the exponent and the base of the logarithm must be the same. Imagine you have \[ 2^{\log_2(8)} \]. According to the identity, this simplifies straight to 8. This identity is very handy because it can dramatically simplify what looks like a complex expression. When you understand this identity, you realize how potent logarithms and exponents can be in simplifying mathematical problems.
Simplifying Expressions
Simplifying expressions often involves recognizing patterns or properties that allow you to condense a complex expression into a simpler form. Using the logarithmic identity is one such method. For instance, take the exercise \[ 4^{\log_4(a-c)} \]. At first glance, it may look complicated, but once you recognize that it fits the form \[ b^{\log_b(x)} \], you can simplify it quickly to \[ a - c \]. Always look for identities or properties that you can apply to make your work easier. You've already reduced a seemingly cumbersome expression into a much simpler form. This makes computation and further algebraic manipulation far more manageable.
Properties of Logarithms
Logarithms have several properties that make them exceptionally useful for solving equations and simplifying expressions. Here are some key properties:
  • Product Rule: \[ \log_b(MN) = \log_b(M) + \log_b(N) \]
  • Quotient Rule: \[ \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \]
  • Power Rule: \[ \log_b(M^k) = k\log_b(M) \]
  • Change of Base Formula: \[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \] where \( c \) is any positive value.
Applying these properties can transform complex logarithmic expressions into simpler ones. For example, if you have an expression like \[ \log_2(32) \], recognizing that \[ 32 = 2^5 \] allows you to use the power rule to simplify it to \[ 5\log_2(2) \], which is just 5. Understanding these properties will make tackling logarithmic and exponential problems much more manageable.

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Most popular questions from this chapter

Prove the power property of logarithms: \(\log _{b} x^{p}=p \log _{b} x\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(3^{6 x+5}=5^{2 x}\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-6 e^{x}-16=0\)

Solve for the indicated variable. \(A=P e^{r t}\) for \(r\) (used in finance)

Compare the graphs of \(Y_{1}=\frac{e^{x}-e^{-x}}{2}\), \(\mathrm{Y}_{2}=\ln \left(x+\sqrt{x^{2}+1}\right)\), and \(\mathrm{Y}_{3}=x\) on the viewing window [-15.1,15.1,1] by \([-10,10,1] .\) Based on the graphs, how do you suspect that the functions are related?
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