Problem 15
Question
Solve the equation for \(x\). $$3^{3 x-4}=3^{5}$$
Step-by-Step Solution
Verified Answer
The short answer to solving the equation \(3^{3x-4} = 3^{5}\) is to equate the exponents, since both sides have the same base of 3: \(3x - 4 = 5\). Solve for x by adding 4 to both sides, giving \(3x = 9\), and then dividing by 3, resulting in \(x = 3\).
1Step 1: Identify the equal base
Notice that both sides of the equation have a base of 3, which means we can equate their exponents.
Step 2: Equate the exponents
2Step 2: Equate the exponents
Since both sides have the same base, we can safely equate the exponents: \(3x - 4 = 5\).
Step 3: Solve the equation for x
3Step 3: Solve the equation for x
Let's solve the equation to find x:
\(3x - 4 = 5\)
Add 4 to both sides:
\(3x = 9\)
Divide both sides by 3:
\(x = 3\)
So, the solution for the equation \(3^{3x-4} = 3^{5}\) is \(x = 3\).
Key Concepts
Exponential FunctionsExponent LawsEquations with Equal Bases
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The general form of an exponential function is \(y = a^x\), where \(a\) is the base and \(x\) is the exponent. When dealing with exponential functions, one key characteristic is that the growth rate is proportional to its current value, making them perfect for modeling phenomena that grow or decay at rates that are proportional to their size, such as populations or investments.
In our example, \(3^{3x - 4}\) is an exponential function with a base of 3 and a variable exponent \(3x - 4\). Since exponential functions can be used to solve various real-world problems, understanding how to work with them can be remarkably beneficial. For instance, in terms of growth, if a population triples every year, the function describing that growth would be an exponential function.
In our example, \(3^{3x - 4}\) is an exponential function with a base of 3 and a variable exponent \(3x - 4\). Since exponential functions can be used to solve various real-world problems, understanding how to work with them can be remarkably beneficial. For instance, in terms of growth, if a population triples every year, the function describing that growth would be an exponential function.
Exponent Laws
Exponent laws, also known as the laws of exponents, are a set of rules that describe how operations involving exponents are carried out. These rules make it easier to manipulate and simplify expressions and are essential when solving equations involving exponents. Some key exponent laws include:
\(a^x \cdot a^y = a^{x+y}\) which describes how to multiply powers with the same base; \(\frac{a^x}{a^y} = a^{x-y}\) that demonstrates the division of powers with the same base; and \( (a^x)^y = a^{xy}\), which shows how to deal with a power raised to another power. In our example, knowing that \(3^{3x - 4} = 3^5\) implies \(3x - 4 = 5\) based on the rule that if the bases are the same, the exponents must be equal, which is fundamental in simplifying exponential equations.
\(a^x \cdot a^y = a^{x+y}\) which describes how to multiply powers with the same base; \(\frac{a^x}{a^y} = a^{x-y}\) that demonstrates the division of powers with the same base; and \( (a^x)^y = a^{xy}\), which shows how to deal with a power raised to another power. In our example, knowing that \(3^{3x - 4} = 3^5\) implies \(3x - 4 = 5\) based on the rule that if the bases are the same, the exponents must be equal, which is fundamental in simplifying exponential equations.
Equations with Equal Bases
When you encounter equations with equal bases on either side of the equation, the process of solving becomes streamlined. Since the bases are the same, you can use the property that equates their exponents, as we did in the example \(3^{3x - 4} = 3^5\). This strategy is built upon the understanding that if \(a^m = a^n\) then \(m=n\), provided that the base \(a\) is not zero or one.
In real-life applications, situations where this property is used include financial calculations involving compound interest or physics problems dealing with exponential decay. By setting the exponents equal to each other, you can focus on solving a typically simpler algebraic equation, rather than dealing directly with the exponential expressions themselves.
In real-life applications, situations where this property is used include financial calculations involving compound interest or physics problems dealing with exponential decay. By setting the exponents equal to each other, you can focus on solving a typically simpler algebraic equation, rather than dealing directly with the exponential expressions themselves.
Other exercises in this chapter
Problem 14
Solve the equation for \(x\). $$5^{-x}=5^{3}$$
View solution Problem 15
According to data obtained from the CBO, the total federal debt (in trillions of dollars) from 2001 through 2006 is given by $$ f(t)=5.37 e^{0.07 s t} \quad(1 \
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Metro Department Store found that \(t\) wk after the end of a sales promotion the volume of sales was given by $$ S(t)=B+A e^{-\lambda t} \quad(0 \leq t \leq 4)
View solution Problem 16
Given that \(\log 3 \approx 0.4771\) and \(\log 4 \approx\) 0.6021, find the value of each logarithm. $$\log \frac{1}{300}$$
View solution