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Chapter 5: Problem 52

Convert to an exponential equation. \(\log _{t} Q=k\)

Short Answer

Expert verified
The exponential form is \(t^k = Q\).

Step by step solution

01

Understand the Logarithmic Form

The given equation is in the form \(\text{log}_t Q = k\). This states that the logarithm of \(Q\) with base \(t\) is equal to \(k\).
02

Recall the Logarithm-Exponential Relationship

The logarithmic equation \(\text{log}_b a = c\) can be rewritten in exponential form as \(b^c = a\).
03

Rewrite the Equation in Exponential Form

Using the rule from the previous step, the base \(t\) raised to the power \(k\) equals \(Q\). Therefore, the equation \(\text{log}_t Q = k\) can be written as \(t^k = Q\).

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Key Concepts

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

Understanding the Logarithmic Form
Logarithms are a way to express exponents. When you see an equation like \(\text{log}_t Q = k\), it means that we are looking for the exponent to which the base \(\text{t}\) must be raised to get \(\text{Q}\).

This form states that \(\text{k}\) is the power you raise \(\text{t}\) to in order to obtain \(\text{Q}\). It's much like saying \(\text{'t raised to the power of k is Q'}\).

This format is very compact, making it easier to manipulate larger numbers and solve for unknown exponents.
The Logarithm-Exponential Relationship
Logarithms and exponentials are two sides of the same coin.

If you have a logarithmic equation like \(\text{log}_b a = c\), you can express it equivalently in exponential form as \(b^c = a\).

This means that logarithms essentially 'invert' exponentiation. If \(\text{log}_b a = c\), then raising \(\text{b}\) to \(\text{c}\) will give you \(\text{a}\).
  • For example, \(\text{log}_2 8 = 3\) because raising 2 to the power of 3 gives 8, i.e., \(2^3 = 8\).
  • Understanding this relationship helps in switching between these two forms, simplifying maths problems significantly.
Converting Logarithms to Exponentials
To convert a logarithmic equation to exponential form, just remember that \(\text{log}_b a = c\) is the same as \(b^c = a\). Let's put this into practice using our initial equation \(\text{log}_t Q = k\).

Here, our base is \(\text{t}\), the argument is \(\text{Q}\), and the exponent is \(\text{k}\). So, translating this into exponential form gives us:

\[t^k = Q\]

This tells us that raising \(\text{t}\) to the power \(\text{k}\) yields \(\text{Q}\). Practicing converting between these two forms will improve your algebraic skills and make solving equations much easier.

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Most popular questions from this chapter

Charitable Giving. Over the last four decades, the amount of charitable giving in the United States has grown exponentially from approximately \(\$ 20.7\) billion in 1969 to approximately \(\$ 316.2\) billion in 2012 (Sources: Giving USA Foundation; Volunteering in America by the Corporation for National \& Community Service; National Philanthropic Trust; School of Philanthropy, Indiana University Purdue University Indianapolis). The exponential function $$ G(x)=20.7(1.066)^{x} $$ where \(x\) is the number of years after \(1969,\) can be used to estimate the amount of charitable giving, in billions of dollars, in a given year. Find the amount of charitable giving in \(1982,\) in \(1995,\) and in \(2010 .\) Then use this function to estimate the amount of charitable giving in \(2017 .\) Round to the nearest billion dollars. (IMAGE CANT COPY)

In Exercises \(77-80\) : a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or a minimum value and find that value.[ 3.3] $$g(x)=x^{2}-6$$

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\log _{a} M-\log _{a} N=\log _{a} \frac{M}{N}$$

E-Cigarette SE-Cigarette Sales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) millionales. The electronic cigarette was launched in 2007 , and since then sales have increased from about \(\$ 20\) million in 2008 to about \(\$ 500\) million in 2012 (Sources: UBS; forbes, \(\mathrm{com}\) ). The exponential function $$ S(x)=20.913(2.236)^{x} $$ where \(x\) is the number of years after \(2008,\) models the sales, in millions of dollars. Use this function to estimate the sales of e-cigarettes in 2011 and in 2015 . Round to the nearest million dollars.

Use a graphing calculator to find the approximate solutions of the equation. $$4 x-3^{x}=-6$$
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