Problem 58 - Algebra A Combined Function

Question

Solve \(5^{x}=9\) by taking the common logarithm of both sides of the equation. Next, solve this equation by taking the natural logarithm of both sides. Compare your solutions. Are they the same? Why or why not?

Step-by-Step Solution

Answer

The solutions are the same; both methods yield \(x \approx 1.365\) because the base change does not affect the value of the logarithmic fraction.

Step 1: Take the Common Logarithm of Both Sides

Start with the equation \(5^x = 9\). Apply the common logarithm (base 10) to both sides to make the equation more manageable. This gives:\[\log(5^x) = \log(9)\]Use the logarithmic identity \(\log(a^b) = b \cdot \log(a)\) to simplify the left side:\[x \cdot \log(5) = \log(9)\]

Step 2: Solve for x Using Common Logarithm

Isolate \(x\) by dividing both sides of the equation by \(\log(5)\):\[x = \frac{\log(9)}{\log(5)}\]Calculate this value using a calculator to find the numerical approximation of \(x\).

Step 3: Take the Natural Logarithm of Both Sides

Again start with the equation \(5^x = 9\). This time, apply the natural logarithm (base \(e\)) to both sides:\[\ln(5^x) = \ln(9)\]Use the logarithmic identity \(\ln(a^b) = b \cdot \ln(a)\) to rewrite the equation as:\[x \cdot \ln(5) = \ln(9)\]

Step 4: Solve for x Using Natural Logarithm

Isolate \(x\) by dividing both sides of the equation by \(\ln(5)\):\[x = \frac{\ln(9)}{\ln(5)}\]Calculate this value using a calculator to determine the numerical approximation of \(x\).

Step 5: Compare the Solutions

Both equations, \(x = \frac{\log(9)}{\log(5)}\) and \(x = \frac{\ln(9)}{\ln(5)}\), provide the same value for \(x\). This is because of the logarithmic property that allows the changing of base without affecting the result due to the constant conversion factor between \(\ln\) and \(\log\). Thus, despite using different bases, the solutions are numerically equal.

Key Concepts

Common Logarithm

A common logarithm is a logarithm with base 10. It is often written simply as \(\log \), without a base written explicitly. Using a common logarithm is especially convenient in applications involving decimal systems, as it simplifies calculations that deal with factors of ten.
When you have an equation like \(5^x = 9\) and apply a common logarithm, you rewrite it as \(\log(5^x) = \log(9)\).
The power rule of logarithms, which states \( \log(a^b) = b \cdot \log(a) \), helps in simplifying this to \(x \cdot \log(5) = \log(9)\). Solving for \(x\) involves dividing both sides by \(\log(5)\) giving \(x = \frac{\log(9)}{\log(5)}\).
Using a calculator, you substitute values to find a numerical solution. Common logarithms offer a clear method for solving exponential equations when working within a base-10 system.

Natural Logarithm

The natural logarithm is denoted by \( \ln \), and it uses base \(e\), where \(e\) is an irrational constant approximately equating to 2.71828.
Natural logarithms are beneficial in many scientific and mathematical contexts due to their natural appearance in calculus and complex systems.
Let's see how to use it to solve \(5^x = 9\). Apply \(\ln\) to get \(\ln(5^x) = \ln(9)\).
The identity \( \ln(a^b) = b \cdot \ln(a) \) lets you simplify this to \(x \cdot \ln(5) = \ln(9)\). Further solving for \(x\), you divide both sides by \(\ln(5)\), resulting in \(x = \frac{\ln(9)}{\ln(5)}\).
Calculating with \(\ln\) involves similar operations as \(\log\), and sometimes calculators label \( \ln \) and \( \log \) for base \(e\) and 10, respectively. The final value obtained using natural logarithms mirrors that of common logarithms, demonstrating their equivalence in generating solutions.

Logarithmic Identity

Logarithmic identities are crucial tools that simplify complex equations. One of the most useful identities is \( \log(a^b) = b \cdot \log(a) \).
This allows the exponent \(b\) to be brought down in front, transforming the logarithm of a power into a simple multiplication operation. This identity works for both common logarithms, \( \log \), and natural logarithms, \( \ln \).
In the problem \(5^x = 9\), using this identity in both bases allows us to transform the equation into a linear form that is easier to handle: \(x \cdot \log(5) = \log(9)\) and \(x \cdot \ln(5) = \ln(9)\).

• Both methods yield equivalent results for \(x\).
• This demonstrates the seamless interchangeability of logarithm bases when solving for unknowns.
• These mathematical properties prove themselves over various types of logarithms, fortifying solutions to exponential equations.

Understanding and applying logarithmic identities effectively is foundational to mastering logarithmic equations.