Problem 4 - Precalculus

Question

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\left(\frac{1}{3}\right)^{-2}=9\)

Step-by-Step Solution

Answer

The logarithmic form of the exponential equation \(\left(\frac{1}{3}\right)^{-2}=9\) is \(\log_{\frac{1}{3}}(9)=-2\).

Step 1: Understand the property

The given property from Theorem 6.2 states that an exponential equation \(b^{a}=c\) can be rewritten as a logarithmic equation \(\log_{b}(c)=a\). Our task is to use this property to rewrite the given equation in its logarithmic form.

Step 2: Identify components of the equation

For the given exponential equation \(\left(\frac{1}{3}\right)^{-2}=9\), identify \(b\), \(a\), and \(c\) from the exponential form. Here, \(b=\frac{1}{3}\), \(a=-2\), and \(c=9\).

Step 3: Apply the property to rewrite the equation

Use the property \(b^{a}=c\) if and only if \(\log_{b}(c)=a\) to rewrite the equation. Substitute \(b=\frac{1}{3}\), \(c=9\), and solve the logarithmic equation \(\log_{\frac{1}{3}}(9)=-2\).

Step 4: Verify the new equation form

Make sure the transformation is correct by checking that the new logarithmic form has equivalent meaning to the original exponential equation. Calculate \(\log_{\frac{1}{3}}(9)\) to ensure it equals \(-2\).

Key Concepts

Theorem 6.2

Theorem 6.2 is a powerful tool in understanding the relationship between exponential and logarithmic equations. This theorem states that any exponential expression of the form \( b^{a} = c \) is equivalent to the logarithmic form \( \log_{b}(c) = a \). Essentially, it allows us to switch between these two forms seamlessly. In an exponential equation, \( b \) is known as the base, \( a \) is the exponent or power, and \( c \) is the result of raising \( b \) to the power \( a \). The theorem enables us to express the same relationship in terms of logarithms, where \( \log_{b}(c) = a \) denotes that \( a \) is the power to which the base \( b \) must be raised to obtain \( c \). This conversion is useful because logarithmic expressions can often simplify complex calculations, especially when solving equations involving exponential growth or decay.

exponential form

Exponential form is a way of expressing numbers using a base and an exponent. In the equation \( b^{a} = c \), \( b \) is the base and \( a \) is the exponent. The result \( c \) is what you get when you multiply the base \( b \) by itself \( a \) times. For example, in \( (\frac{1}{3})^{-2} = 9 \), \( (\frac{1}{3}) \) is the base and \(-2\) is the exponent. Raising \( (\frac{1}{3}) \) to the power of \(-2\) effectively means taking the reciprocal squared, yielding \( 9 \).

• The base determines the "size" of the increments.
• The exponent indicates how many times to use the base as a factor.
• A negative exponent implies taking the reciprocal.

Understanding exponential form is crucial because it serves as a basis for transforming into logarithmic form, utilizing Theorem 6.2.

logarithmic form

Logarithmic form is a way to express an exponential relationship in terms of logarithms. In the context of Theorem 6.2, if you have an exponential equation \( b^{a}=c \), it can be rewritten in logarithmic form as \( \log_{b}(c)=a \). This transformation is exceptionally useful for solving equations where the exponent is unknown.

• The base \( b \) of the logarithm is the same as the base of the exponential equation.
• The result \( c \) becomes the input to the logarithm.
• The exponent \( a \) is what the logarithm equals.

So in the example \( (\frac{1}{3})^{-2}=9 \), transforming it using the log form becomes \( \log_{(\frac{1}{3})}(9) = -2 \). This process provides an alternate method to verify or solve for unknown values in equations involving powers.

properties of logarithms

The properties of logarithms are instrumental in manipulating and simplifying logarithmic expressions. These include basic rules that can make complex calculations easier and more intuitive. Here are some key properties:

• Product Rule: \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \) allows us to break down the log of a product into a sum.
• Quotient Rule: \( \log_{b}(M/N) = \log_{b}(M) - \log_{b}(N) \) helps split a division inside a logarithm into a difference.
• Power Rule: \( \log_{b}(M^n) = n \cdot \log_{b}(M) \) simplifies the log of a power into a multiplication.
• Change of Base Formula: \( \log_{b}(c) = \frac{\log_{k}(c)}{\log_{k}(b)} \) where \( k \) is a new base, helpful for computing logs on calculators.

Understanding these properties gives you a toolkit for translating problems and finding solutions with more ease and clarity. They underpin many algebraic manipulations in more advanced mathematics.