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Chapter 13: Problem 36

Determine the exact value of each of the given expressions. $$10^{2 \log _{10} 3}$$

Short Answer

Expert verified
The exact value is 9.

Step by step solution

01

Understanding the Exponent

The expression is given as \( 10^{2 \log_{10} 3} \). Here, \( 2 \log_{10} 3 \) is acting as an exponent. We can interpret this exponent using properties of logarithms.
02

Applying Logarithm Power Rule

According to logarithm rules, \( n \log_b a = \log_b a^n \). So, we can rewrite the exponent \( 2 \log_{10} 3 \) as \( \log_{10} (3^2) \). This transforms the original expression to \( 10^{\log_{10} 3^2} \).
03

Using Inverse Properties of Logs and Exponents

Recall that if a base and its logarithm power are the same, they cancel each other out due to the property \( b^{\log_b a} = a \). So, \( 10^{\log_{10} 3^2} = 3^2 \).
04

Calculating the Final Value

Now calculate \( 3^2 \). Multiplying 3 by itself gives \( 3 \times 3 = 9 \). Therefore, the expression evaluates to 9.

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Key Concepts

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

Logarithm Properties
Understanding logarithms is key to mastering exponential functions. A logarithm answers the question: "To what power must we raise a base to get a certain number?" For example, in the equation \( \log_{10} a = b \), 10 is the base and \( b \) is the power; thus, \( 10^b = a \). Here are some basic properties of logarithms that are widely used:
  • **Product Rule:** \( \log_b (xy) = \log_b x + \log_b y \)
  • **Quotient Rule:** \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \)
  • **Change of Base Formula:** \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is any positive value
  • **Logarithm of One:** The log of one for any base is zero, \( \log_b 1 = 0 \).
Applying these properties helps break down complex expressions into manageable calculations, simplifying exponential equations like \( 10^{2 \log_{10} 3} \).
Power Rule of Logarithms
An essential tool in simplifying logarithmic expressions is the Power Rule. It states that an exponent within a logarithm can be moved as a coefficient: \( n \log_b a = \log_b a^n \).
This property allows you to take any power within the log function and bring it to the front as a multiplication. Let's apply it:
  • Given the expression \( 2 \log_{10} 3 \), use the power rule to rewrite it as \( \log_{10} (3^2) \).
This transformation is key to simplifying complex expressions by turning a logarithm multiplied by a number into a logarithm of a power. Understanding and using the power rule effectively can help solve problems much faster.
Inverse Properties of Logarithms
Inverse properties of logarithms are powerful tools for simplifying expressions where logs and exponents are involved. Particularly, the property \( b^{\log_b a} = a \) is crucial.
This property reflects the fact that exponential and logarithmic functions are inverses of each other:
  • If you have logs and the base of the exponent (like \( 10^{\log_{10} x} \)), they cancel out the same way a function and its inverse do, leaving just \( x \).
Applying this principle simplifies the expression \( 10^{\log_{10} 3^2} \) to \( 3^2 \). Thus, expressions wrapped in inverse operations can be dramatically simplified by canceling the inverse components.

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Most popular questions from this chapter

Find the natural antilogarithms of the given logarithms. $$-0.7429$$

\(\text {Solve the given problems.}\) A study of urban density shows that the population density \(D\) (in persons/mi \(^{2}\) ) is related to the distance \(r\) (in mi) from the city center by \(\log _{e} D=\log _{e} a-b r+c r^{2},\) where \(a, b,\) and \(c\) are positive constants. Solve for \(D\) as a function of \(r\)

$$\text {Plot the indicated graphs.}$$ On the moon, the distance \(s\) (in \(\mathrm{ft}\) ) a rock will fall due to gravity is \(s=2.66 t^{2},\) Where \(t\) is the time (in s) of fall. Plot the graph of \(s\) as a function of \(t\) for \(0 \leq t \leq 10\) s on (a) a regular rectangular coordinate system and (b) a semilogarithmic coordinate system.

Perform the indicated operations. The time \(t\) (in ps) required for \(N\) calculations by a certain computer design is \(t=N+\log _{2} N .\) Sketch the graph of this function.

Use a calculator to solve the given equations. What values of \(x\) cannot be solutions of the equation \(y=\log (2 x-5)+\log \left(x^{2}+1\right) ?\)
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