Question

Solve \(6^{x}=5^{x+1}\). Express the answer both in exact form and as a decimal rounded to three decimal places.

Short answer

Exact: \( x = \frac{\ln(5)}{\ln(6) - \ln(5)} \). Decimal: \( x \approx 8.791 \).

Step-by-step solution

Take the Natural Logarithm of Both Sides

Apply the natural logarithm on both sides of the equation: \[ \ln(6^x) = \ln(5^{x+1}) \]

Use Logarithm Properties to Simplify

Utilize the power rule of logarithms, \( \ln(a^b) = b \ln(a) \), to simplify the equation: \[ x \ln(6) = (x+1) \ln(5) \]

Distribute the Logarithm on the Right Side

Expand the right side: \[ x \ln(6) = x \ln(5) + \ln(5) \]

Isolate the Variable

Move all terms involving \( x \) to one side: \[ x \ln(6) - x \ln(5) = \ln(5) \] Factor out \( x \): \[ x(\ln(6) - \ln(5)) = \ln(5) \]

Solve for x

Isolate \( x \) by dividing by \( (\ln(6) - \ln(5)) \): \[ x = \frac{\ln(5)}{\ln(6) - \ln(5)} \] Using a calculator, evaluate the expression: \[ x \approx \frac{1.609}{1.792-1.609} \approx \frac{1.609}{0.183} \approx 8.791 \]

Key concepts

Natural Logarithm

The natural logarithm (denoted as \(\text{ln}\)) is a special type of logarithm with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are often used in mathematics, especially in solving exponential equations because of their unique properties.

When solving exponential equations like \(6^x = 5^{x+1}\), taking the natural logarithm of both sides simplifies the process. For instance, applying the natural logarithm to \(6^x\) gives \(\text{ln}(6^x)\), which using the power rule of logarithms becomes \(x \text{ln}(6)\). This simplification makes it easier to isolate and solve for the variable \(x\).

Natural logarithms are instrumental in many fields such as biology, economics, and physics due to their relationship with exponential growth and decay. For this exercise, understanding how to apply the natural logarithm is crucial to break down and simplify the given equation.

Logarithm Properties

Logarithm properties are critical tools that help simplify complex exponential equations. Here are some key properties that you should know:

• Product Rule: \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \) – This property states that the logarithm of a product is the sum of the logarithms of its factors.


• Quotient Rule: \( \text{ln}\frac{a}{b} = \text{ln}(a) - \text{ln}(b) \) – This property states that the logarithm of a quotient is the difference of the logarithms.


• Power Rule: \( \text{ln}(a^b) = b \text{ln}(a) \) – This property is particularly useful in our problem because it allows us to bring down the exponent as a coefficient.

In our exercise, we used the power rule twice to simplify both sides of the equation: \( \text{ln}(6^x) = \text{ln}(5^{x+1}) \) becomes \( x \text{ln}(6) = (x+1) \text{ln}(5) \). Then, by distributing and isolating the variable, we leveraged these properties to make the equation more manageable. Understanding and applying these logarithm properties are essential to solving exponential equations effectively.

Solving Equations

Solving equations involves finding the value(s) of the variable that make the equation true. For exponential equations, such as \(6^x = 5^{x+1}\), we often employ logarithms to simplify and solve for the variable. Here is a step-by-step breakdown of the process used in the exercise:

• Step 1: Take the natural logarithm of both sides: \(\text{ln}(6^x) = \text{ln}(5^{x+1})\).


• Step 2: Apply the power rule of logarithms: \(\text{ln}(6^x) = x \text{ln}(6)\) and \(\text{ln}(5^{x+1}) = (x+1) \text{ln}(5)\).


• Step 3: Expand the equation: \(x \text{ln}(6) = x \text{ln}(5) + \text{ln}(5) \).


• Step 4: Isolate the variable \(x\) by moving terms: \(x \text{ln}(6) - x \text{ln}(5) = \text{ln}(5)\) and factor out \(x\): \(x(\text{ln}(6) - \text{ln}(5)) = \text{ln}(5)\).


• Step 5: Solve for \(x\) by dividing both sides by \(\text{ln}(6) - \text{ln}(5)\): \( x = \frac{\text{ln}(5)}{\text{ln}(6) - \text{ln}(5)} \).

Finally, to get a decimal approximation, we use a calculator to compute: \(\text{ln}(5) \approx 1.609\), \(\text{ln}(6) \approx 1.792\) leading to \( x \approx 8.791 \). This full process showcases how logarithms can simplify and solve exponential equations effectively.