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Chapter 8: Problem 43

Solve each system of exponential or logarithmic equations. $$\begin{aligned} &y=3^{x}\\\ &y=2^{x} \end{aligned}$$

Short Answer

Expert verified
x = 0.

Step by step solution

01

Set Equations Equal

Given the system of equations:y = 3^xy = 2^xSince both expressions are equal to y, set them equal to each other:3^x = 2^x
02

Take the Logarithm of Both Sides

Apply logarithms to both sides of the equation. Using the natural logarithm (ln) for convenience:\( \text{ln}(3^x) = \text{ln}(2^x) \)
03

Use Logarithmic Properties

Apply the power rule of logarithms, which states \( \text{ln}(a^b) = b \text{ln}(a) \):\( x \text{ln}(3) = x \text{ln}(2) \)
04

Simplify the Equation

Divide both sides of the equation by \( x \) to isolate the natural logarithms:\( \text{ln}(3) = \text{ln}(2) \)
05

Solve for x

Since \( \text{ln}(3) \) and \( \text{ln}(2) \) are distinct values, the only solution satisfying this equality is:\( x = 0 \)
06

Verify the Solution

To verify, substitute \( x = 0 \) back into the original equations:\( y = 3^0 = 1 \)\( y = 2^0 = 1 \)Both verifications give y = 1, so the solution is confirmed.

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Key Concepts

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

System of Equations
A system of equations consists of two or more equations with the same variables. In our case, we had the equations:
  • y = 3^x
  • y = 2^x
To find a solution, we look for values of the variables that satisfy all equations simultaneously. Here, this means determining values of `x` and `y` where both equations are true simultaneously. Setting the equations equal, since they both equal `y`, is a key step in solving the system. This transforms our problem into handling an exponential equation by comparing `3^x` to `2^x`.
Natural Logarithm
The natural logarithm, denoted as `ln`, is a logarithm with base `e`, where `e 鈮� 2.71828`. It's a very useful function because of its properties, especially in calculus and solving exponential equations.
When we have `3^x = 2^x`, applying the natural logarithm on both sides converts the exponential form into a linear form. This is why we write:
  • ln(3^x) = ln(2^x)
Using the natural log helps simplify the process by leveraging its properties to make the equation more manageable.
Logarithmic Properties
Logarithmic properties help in transforming and simplifying equations involving exponents. Two important properties we used here are:
  • Power Rule: ln(a^b) = b * ln(a)
  • Log of a Product: ln(a * b) = ln(a) + ln(b)
For our particular problem, we used the power rule to bring down the exponent `x` as a coefficient:
  • ln(3^x) = x * ln(3)
  • ln(2^x) = x * ln(2)
This simplification is crucial in isolating `x` and solving the equation. By applying logarithmic properties, complex exponential equations can be converted to simpler algebraic forms.
Power Rule of Logarithms
The power rule of logarithms states that for any numbers `a` and `b`, ln(a^b) = b * ln(a). This rule is particularly useful for dealing with exponents in logarithmic equations. In our problem:
  • ln(3^x) becomes x * ln(3)
  • ln(2^x) becomes x * ln(2)
By rewriting the equation using the power rule, it transitions from an exponential equation (`3^x = 2^x`) to an equation that is linear in terms of `x` (`x * ln(3) = x * ln(2)`). This makes it easier to isolate `x`.
When we simplify further, we see that dividing both sides by `x` (as long as `x` 鈮� 0) gives us `ln(3) = ln(2)`, which is not true. So, the only valid solution is `x = 0`, confirming our initial setup.

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Most popular questions from this chapter

A system of equations can be used to find the equation of a line that goes through two points. For example, if \(y=a x+b\) goes through \((3,5),\) then a and b must satisfy \(3 a+b=5 .\) For each given pair of points, find the equation of the line \(y=a x+b\) that goes through the points by solving a system of equations. $$(-2,3),(4,-7)$$

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Solve the system of equations \(2 x-3 y=20\) and \(x+4 y=-1\)

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