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Chapter 2: Problem 15

(a) \(\log _{b} a=c\). (b) \(a=b^{c}\).

Short Answer

Expert verified
Convert \log_{b} a = c\ to exponential form: a = b^c.

Step by step solution

01

- Understanding Logarithms

Recall that \(\text{logarithms}\) are the inverses of exponentiation. Specifically, if \(y = \log_b x\), then \(b^y = x\). This means that \(\text{logarithm}\) \(c \) is the \(\text{power}\) to which base \(b \) must be raised to get \(a \).
02

- Given Logarithmic Form

Consider the given logarithmic equation: \( \log_{b} a = c \). This represents the exponent form.
03

- Converting to Exponential Form

To convert the logarithmic equation to an exponential form, use the inverse property of logarithms: \( b^{\log_b a} = a \. \Therefore, \ a = b^c\).
04

- Verification

Check the given exponential equation: \(a = b^c \. \This matches our derived result,confirming that \ \log_b a = c \) is correct.

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Key Concepts

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

logarithmic equation
A logarithmic equation links a number to its logarithm. Put simply, a logarithm answers the question: 'To what power must we raise the base to get this number?'

In the exercise, we have: \( \log_b a = c \ \), which means:
\( \c \) is the power to which \( \b \) must be raised to get \( \a \).

Logarithmic equations can help solve problems where you need to find the exponent. Here are the steps to solve them:
  • Identify the base \( \( b \) \).
  • Identify the argument \( a \).
  • Identify the logarithm (result) \( c \).

Understanding logarithms is crucial for solving exponential equations and many real-world problems, like computing interest rates or studying population growth.
exponential form
The exponential form is the inverse of the logarithmic form. Here, instead of asking 'What power should the base be raised to?', you're stating 'The base raised to a specific power equals the number'.

In the exercise, we convert: \( \log_b a = c \) to \( \a = b^c \).

To do this, just remember:
  • The base \( b \) remains the same.
  • The logarithm result \( c \) becomes the exponent.
  • The argument \( a \) becomes the result.

Converting to exponential form can make it easier to solve for unknowns or calculate specific values. It directly shows how bases and exponents are connected.
inverse properties
Logarithms and exponents are inverses of each other. This means they undo each other鈥檚 functions.

Here鈥檚 what it means: If \( \y = \log_b x \, \b^y = x \).

Here鈥檚 an example:
  • If \( \10^2 = 100 \), then \( \log_{10} 100 = 2 \).

In our exercise, we see the inverse relationship. The logarithmic form \( \log_b a = c \) can be converted to \( \a = b^c \). Notice how \( \log \) and \( exponentiation \) work oppositely to find and verify values.

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Most popular questions from this chapter

If \(2^{2 x+1}-6\left(2^{x}\right)=0\) then \(x\) is: (a) \(1.5\) (b) \(\log _{2} 3\) (c) \(\log 3\) (d) \(\log _{3} 2\) (e) 3 .

In the expansion of \((a-2 b)^{3}\) the coefficient of \(b^{2}\) is: (a) \(-2 a^{2}\) (b) \(-8 a\) (c) \(12 a\) (d) \(-4 a\) (e) \(-12\).

Express \(\log _{9} x y\) in terms of \(\log _{3} x\) and \(\log _{3} y\). Without using tables, solve for \(x\) and \(y\) the simultaneous equations $$ \begin{gathered} \log _{9} x y=\frac{5}{2} \\ \log _{3} x \log _{3} y=-6 \end{gathered} $$ expressing your answers as simply as possible.

\(3 \log x+1=\log 10 x^{3}\) is an equation.

In the expansion of \((1+x)^{6}\) the coefficient of \(x\) is 6 .
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