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Chapter 10: Problem 24

Write in exponential form. $$\log _{64} 2=\frac{1}{6}$$

Short Answer

Expert verified
The exponential form is \ \( 64^{\frac{1}{6}} = 2 \).

Step by step solution

01

Understand the Logarithm Equation

The given logarithmic equation is \ \ \( \log _{64} 2=\frac{1}{6} \) \ \ This means that the base 64 raised to some exponent equals 2. In logarithmic terms, \ \( \log_b(a)=c \) can be rewritten in exponential form as \ \( b^c = a \). \ \ Here, \( b = 64 \), \( a = 2 \) and \( c = \frac{1}{6} \).
02

Rewrite the Equation in Exponential Form

From \ \( \log_{64} 2 = \frac{1}{6} \), convert it to its exponential form using the property \ \( b^c = a \): \ \ \( 64^{\frac{1}{6}} = 2 \).

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Key Concepts

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

Logarithmic Equations
Logarithmic equations involve the logarithmic function, which is the inverse of an exponential function. In a logarithmic equation, such as \( \log_{64}{2} = \frac{1}{6} \), we are asked to find the power to which the base (64) must be raised to get the number (2). Here: \ \
  • 64 is the base
  • 2 is the number we want to achieve (known as the argument)
  • \frac{1}{6} is the exponent, the power to which the base is raised to get the argument
Logarithmic equations can often look complicated, but they essentially ask the same question as their exponential equivalents. Understanding this inverse relationship is key to solving logarithmic problems.
Exponential Equations
Exponential equations are equations in which variables appear as exponents. An example of an exponential form is the solution to the logarithmic problem given: \
  • \(64^{\frac{1}{6}} = 2\)
This signifies that 64 raised to the power of 1/6 equals 2. To understand this: \
  • The base (here 64) is the number that is raised to a power
  • The exponent (here 1/6) shows how many times the base is multiplied by itself
Exponential equations appear frequently in various branches of mathematics, science, and engineering, particularly in growth models, compounding interest, and radioactive decay, where growth or decay is proportional to the current amount at any time.
Conversion Between Logarithmic and Exponential Form
Conversion between logarithmic and exponential forms is a crucial skill. Knowing how to convert between these two forms allows one to switch perspectives and solve equations more efficiently. The general conversion rule follows: \
  • Logarithmic form: \(\log_b{a} = c\)
  • Exponential form: \(\b^c = a\)
To convert from a logarithmic form to an exponential form, recognize the components and apply the rule. In our exercise, \
\(\log_{64}{2} = \frac{1}{6} \) converts to its exponential form: \ \
  • Expression: \log_{64}{2} = \frac{1}{6} \
  • Conversion: \64^{\frac{1}{6}} = 2 \
Converting between these forms not only helps in solving equations but also in understanding their properties and representations thoroughly.

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Most popular questions from this chapter

Graph each logarithmic function. $$g(x)=\log _{1 / 6} x$$

Find the amount of money in an account after 8 yr if \(\$ 4500\) is deposited at \(6 \%\) annual interest compounded as follows. (a) Annually (b) Semiannually (c) Quarterly (d) Daily (Use \(n=365 .)\) (e) Continuously

Suppose that in solving a logarithmic equation having the term \(\log (3-x)\), we obtain the proposed solution -4 . We know that our algebraic work is correct, so we reject -4 and give \(\varnothing\) as the solution set.

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{\ln 2 x}=e^{\ln (x+1)} $$

Solve each equation. Give exact solutions. $$ \log _{4}(2 x+8)=2 $$
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