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Chapter 4: Problem 21

solve for \(x\) without using a calculating utility. $$\log _{5}\left(5^{2 x}\right)=8$$

Short Answer

Expert verified
x = 4

Step by step solution

01

Apply the Logarithm Property

The logarithm property \(\log_b(b^y) = y\) can be applied here. Since we have \(\log_{5}(5^{2x})\), we can simplify this expression by moving \(2x\) out of the logarithm, giving us \(2x = \log_5(5^{2x})\).
02

Equate the Simplified Expression to 8

With the expression simplified to \(2x = 8\), we can now solve for \(x\).
03

Solve for \(x\)

Divide both sides of the equation \(2x = 8\) by 2 to isolate \(x\). This yields \(x = \frac{8}{2} = 4\).

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Key Concepts

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

Understanding Logarithm Properties
Logarithms are powerful tools in mathematics that let us work with exponential relationships easily. One key property is the rule: \( \log_b(b^y) = y \). This property states that if we take the logarithm of a number which is a base raised to a power, we simply get the exponent back. For example:
  • If you have \( \log_{5}(5^{3}) \), it simplifies to \( 3 \).
  • This is because the base of the logarithm (5) matches the base of the exponent (5).
This property is incredibly useful for simplifying complex logarithmic expressions, allowing us to solve equations more efficiently. It also helps transform exponential expressions into linear ones, making them easier to manage in algebraic processes. A solid understanding of this fundamental property is essential for mastering logarithms.
Breaking Down Exponential Equations
Exponential equations are equations in which variables appear as exponents. In our exercise, we have an exponential form inside the logarithmic expression: \( 5^{2x} \). To solve an equation involving an exponent, we often use logarithms to "bring down" the exponent into a more manageable form.
  • This is where the log property \( \log_b(b^y) = y \) plays a vital role.
  • For example, applying \( \log_5 \) to an equation like \( \log_5(5^{2x}) = 8 \) turns it into a solvable linear equation \( 2x = 8 \).
The primary aspect of dealing with exponential equations is understanding how to transition them into simpler forms, often involving logarithms or by directly comparing exponential terms. Familiarity with these transformations unlocks the ability to solve complex problems efficiently.
The Art of Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate desired variables, making it a crucial skill in mathematics. Once you understand the core properties of logarithms and exponents, manipulating these equations becomes straightforward.In our exercise, we simplified \( \log_5(5^{2x}) = 8 \) to \( 2x = 8 \). The next step was to isolate \( x \), which involved dividing both sides by 2:
  • First, begin with the equation \( 2x = 8 \).
  • Next, divide each term by 2, giving \( x = \frac{8}{2} \).
  • Finally, solve to find \( x = 4 \).
This example demonstrates how foundational math skills, such as dividing both sides of an equation, can help find solutions quickly. Mastering algebraic manipulation enables you to approach any mathematical problem with confidence and precision.

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Most popular questions from this chapter

Find \(d y / d t\) in terms of \(x, y,\) and dx/dt, assuming that \(x\) and \(y\) are differentiable functions of the variable \(t .\) [Hint: Differentiate both sides of the given equation with respect to \(t .1\) $$x^{3} y^{2}+y=3$$

In Exercises find \(d y / d x\). $$y=(\ln x)^{2}$$

Find the limit. $$\lim _{x \rightarrow 0^{+}} \frac{\ln (\sin x)}{\ln (\tan x)}$$

simplify the expression without using a calculating utility. (a) \(2^{-4}\) (b) \(4^{1.5}\) (c) \(9^{-0.5}\)

Find the limit. $$\lim _{x \rightarrow 0}(\csc x-1 / x)$$
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