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Chapter 4: Problem 5

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

Short Answer

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

Step by step solution

01

Understand the Given Logarithmic Equation

The logarithm \(\text{log}_a K = J\) indicates that 'J' is the exponent to which the base 'a' must be raised to get 'K'.
02

Convert the Logarithmic Form to Exponential Form

To convert the logarithmic equation \(\text{log}_a K = J\) into its equivalent exponential form, recall the definition of a logarithm. It can be written as \( a^J = K \).
03

Write the Final Equivalent Exponential Equation

From the previous step, the equivalent exponential equation for \(\text{log}_a K = J\) is \(a^J = K\).

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Key Concepts

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

logarithms
A logarithm is a way to solve for an exponent in equations of the form: \(y = log_b(x)\). This equation shows that y is the power you need to raise the base b to in order to get x.
For example: \(log_2(8) = 3\) because \(2^3 = 8\).
Logarithms answer the question: **'To what exponent must the base be raised, to produce a given number?'** They are common in many areas, such as science, engineering, and finance.
exponential form
Exponential form is another way to write equations, involving exponents. It looks like: \(b^y = x\).
This form states that b (base) raised to y (the exponent) equals x.
If you see the equation \(2^3 = 8\), it is in exponential form.
The exponential form is particularly useful in growth models, decay equations, and complex calculations involving powers and roots.
conversion between logarithms and exponents
Converting between logarithmic and exponential forms is a fundamental skill that helps to solve equations more easily.
If you have \(log_b(x) = y\), you convert it into exponential form as \(b^y = x\).
This shows how exponents and logarithms are inversely connected.
Here is a simple step-by-step process:
1. Identify the logarithm equation.
2. Remember that \(log_b(x) = y\) implies \(b^y = x\).
3. Rewrite the equation into exponential form.
For example: if \(log_3(81) = 4\), we can write it as \(3^4 = 81\).
This skill is useful in algebra, calculus, and solving real-world problems!

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Most popular questions from this chapter

Differentiate. $$ f(x)=5 \log x $$

Differentiate. $$ y=3 x^{-2}-\frac{1}{e^{2 x}}+5 \sqrt{x}-2 $$

Show that any two measurements of an exponentially growing population will determine \(k\). That is, show that if \(y\) has the values \(y_{1}\) at \(t_{1}\) and \(y_{2}\) at \(t_{2}\), then $$ k=\frac{\ln \left(y_{2} / y_{1}\right)}{t_{2}-t_{1}} $$

Double Declining-Balance Depreciation. An office machine is purchased for \(\$ 5200 .\) Under certain assumptions, its salvage value \(V\) depreciates according to a method called double declining balance, basically \(80 \%\) each year, and is given by $$ V(t)=\$ 5200(0.80)^{\ell} $$ where \(t\) is the time, in years. a) Find \(V^{\prime}(t)\). b) Interpret the meaning of \(\mathbf{V}^{\prime}(t)\).

The probability that a tadpole with mass \(0.1 \mathrm{~g}\) is captured by notonectids (aquatic insects) in an experiment is $$ P=1-\frac{1}{1+e^{-(0.055+0.083 T)}} $$ where \(T\) is the water temperature in degrees Celsius. Suppose that when the water temperature is \(20^{\circ} \mathrm{C}\), it is rising at the rate of 2 degrees per hour. How fast is the probability of capture decreasing? \(^{6}\)
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