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Chapter 3: Problem 93

In Exercises \(93-102,\) solve each equation. $$5^{2 x} \cdot 5^{4 x}=125$$

Short Answer

Expert verified
The solution to the equation is \(x = 0.5\).

Step by step solution

01

Recognize the base numbers

Observe that all numbers in the equation are powers of 5. Also, 125 can be rewritten as \(5^3\). Rewriting them gives us \(5^{2x} \cdot 5^{4x} = 5^3\).
02

Apply exponent rule

The rule of multiplication in exponents states that when bases are the same, the exponents are added. For the left side of the equation, apply the rule of addition which results in \(5^{(2x+4x)} = 5^3\). This simplifies to \(5^{6x} = 5^3\).
03

Equalize the exponents

Since the bases are equal, the corresponding exponents must also be equal. Hence, setting the exponents equal to each other gives us 6x = 3.
04

Solve for the variable

Finally, divide both sides by 6 to solve for x: \(x = \frac{3}{6} = 0.5\).

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Key Concepts

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

Exponent Rules
Exponent rules make working with powers easier and more intuitive. One important rule to remember is that if you have the same base, you can add or subtract the exponents when you multiply or divide the expressions, respectively. In the exercise given, the rule that is particularly useful is the multiplication rule. This rule states:
  • If you multiply two exponential expressions with the same base, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).
When you apply this rule, the base remains unchanged while only the exponents are manipulated. So, when given an equation such as \(5^{2x} \cdot 5^{4x} \), noticing they share the base of 5 allows you to add the exponents: \(2x + 4x = 6x \). Ensuring mastery of these rules can greatly reduce the complexity of exponential expressions.
Powers of Numbers
Understanding the powers of numbers involves recognizing how numbers can be expressed as multiplied self-factors. For example, the expression \(5^3\) is shorthand for multiplying the number 5 by itself three times: \(5 \times 5 \times 5\).
  • This helps to simplify and solve equations where numbers are expressed as powers.
In the exercise, recognizing that 125 is actually \(5^3\) is crucial. This recognition allows us to rewrite the equation \(5^{2x} \cdot 5^{4x} = 125\) more uniformly as \(5^{6x} = 5^3\). By expressing all components of the equation in terms of the same base, we are set up for easier algebraic manipulation and solution.Understanding these conversions can often be pivotal in breaking down what seems like a complex equation into solvable parts.
Algebraic Manipulation
Algebraic manipulation involves re-arranging and simplifying equations to find the value of variables. Once we have simplified the equation using exponent rules into \(5^{6x} = 5^3\), it is clear that the two sides of the equation must balance.
  • This means that the exponents must be equal since the bases are already equal.
Thus, if \(6x = 3\), then solving for \(x\) involves straightforward steps of division. Divide both sides by 6:
  • \(6x = 3 \)
  • \(x = \frac{3}{6} = 0.5 \)
These steps highlight how algebraic manipulation helps in isolating variables and finding their values. Being methodical in such manipulations ensures accuracy in solutions.

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Most popular questions from this chapter

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

Consider the quadratic function $$ f(x)=-4 x^{2}-16 x+3 $$ a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range.

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?
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