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Chapter 4: Problem 6

$$ \text { Write an equivalent exponential equation. } $$ $$ \log _{a} J=K $$

Short Answer

Expert verified
The equivalent exponential equation is \ a^K = J.\

Step by step solution

01

- Understand the logarithmic form

The logarithmic form given is \ \ \( \log_{a} J = K \). This means that the logarithm of J base a is equal to K.
02

- Convert logarithmic to exponential form

Recognize that any logarithmic equation of the form \ \ \( \log_{b} x = y \) can be rewritten in exponential form as \ \ \( b^y = x \).
03

- Apply the exponential conversion

Using the rule from Step 2, convert \ \ \( \log_{a} J = K \) to exponential form. According to the rule \ \ \( a^K = J \).

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Key Concepts

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

logarithmic equations
Logarithmic equations are equations that involve the logarithm of an expression. A logarithm is essentially the opposite of exponentiation. Instead of asking what is the result if you raise a number to a certain power (like in exponential equations), a logarithm asks what power should you raise a base number to, in order to get another number. For example, in the equation \( \log_{a} J = K \), you're asking: 'To what power must we raise the base 'a' to obtain 'J'?'
exponential equations
Exponential equations are equations where variables appear in the exponent. For example: \( b^y = x \). This means 'b' raised to the power of 'y' equals 'x'. Exponential equations can be transformed from logarithmic equations through a specific conversion method. It鈥檚 often easier to work with exponential forms because they are more straightforward to solve for the variable when compared to their logarithmic counterparts.
In the provided example, \( \log_{a} J = K \) was converted to \( a^K = J \), thus transforming it into an exponential equation.
mathematical conversions
Mathematical conversions are processes for changing one form of an equation or expression into another that is equivalent. Understanding conversions between different forms is crucial in solving complex mathematical problems.
To convert a logarithmic equation to exponential form, remember this rule: If \( \log_{b} x = y \), then the equivalent exponential form is \( b^y = x \).
  • Identify the base (b) from the logarithm.
  • Identify the result of the logarithm (x).
  • Recognize the logarithm's outcome (y) as the exponent.
Using this conversion, the logarithmic equation \( \log_{a} J = K \) was converted to the exponential equation \( a^K = J \).

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Most popular questions from this chapter

Medication Conccntration. The concentration \(C\) in parts per million, of a medication in the body \(t\) hours after ingestion is given by the function $$ C(t)=10 t^{2} e^{-t} $$ a) Find the concentration after \(0 \mathrm{hr} ; \mathrm{l} \mathrm{hr} ; 2 \mathrm{hr} ;\) \(3 \mathrm{hr} ; 10 \mathrm{hr}\). b) Sketch a graph of the function for \(0 \leq t \leq 10\) c) Find the rate of change of the concentration \(C^{\prime}(t) .\) d) Find the maximum value of the concentration and where it occurs. e) Interpret the meaning of the derivative.

Suppose that you are given the task of learning \(100 \%\) of a block of knowledge. Human nature tells us that we would retain only a percentage \(P\) of the knowledge \(t\) weeks after we have learned it. The Ebbinghaus learning model asserts that \(P\) is given by $$ P(t)=Q+(100 \%-Q) e^{-h t} $$ where \(Q\) is the percentage that we would never forget and \(h\) is a constant that depends on the knowledge learned. Suppose that \(Q=40 \%\) and \(k=0.7\) a) Find the percentage retained after \(0 \mathrm{wk}\); 1 wk; 2 wk; 6 wk; 10 wk. b) Find \(\lim _{\ell \rightarrow \infty} P(\iota)\) c) Sketch a graph of \(P\). d) Find the rate of change of \(P\) with respect to time \(t, P^{\prime}(t)\). e) Interpret the meaning of the derivative.

The intensity I of an earthquake is given by $$ I=I_{0} 10^{R} $$ where \(R\) is the magnitude on the Richter scale and Io is the minimum intensity, where \(R=0\), used for comparison. a) Find \(I\), in terms of \(I_{0}\), for an earthquake of magnitude 7 on the Richter scale. b) Find \(I\), in terms of \(I_{0}\), for an earthquake of magnitude 8 on the Richter scale. c) Compare your answers to parts (a) and (b). d) Find the rate of change \(d I / d R\). e) Interpret the meaning of \(d I / d R .\)

The annual death rate (per thousand) in Mexico may be modeled by the function $$ D(t)=34.4-\frac{30.48}{1+29.44 e^{-0.072 t}} $$ where \(t\) is the number of years after \(1900.5\) a) Find \(D^{\prime}(t)\) and interpret its meaning. b) Between 2007 and 2008 , by how much will Mexico's death rate decline?

An amount is invested at \(7.3 \%\) per year compounded continuously. What is the effective annual yield?
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