Solve $9^{x-5}=12^x$. Enter An Exact Answer Or Round Your Answer To The Nearest Tenth. Do Not Include $x=$ In Your Answer.

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Introduction

Solving exponential equations can be a challenging task, especially when dealing with different bases. In this case, we are given the equation $9^{x-5}=12^x$ and we need to find the exact value of $x$. This equation involves two different bases, 9 and 12, and we need to use various techniques to isolate the variable $x$.

Step 1: Take the Logarithm of Both Sides

To solve this equation, we can start by taking the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation. We can use the natural logarithm (ln) or the common logarithm (log) to take the logarithm of both sides.

\log(9^{x-5}) = \log(12^x)

Step 2: Apply the Power Rule of Logarithms

The power rule of logarithms states that $\log_b(M^r) = r\log_b(M)$. We can apply this rule to both sides of the equation to simplify it.

(x-5)\log(9) = x\log(12)

Step 3: Distribute the Logarithm

We can distribute the logarithm on the left side of the equation to simplify it further.

x\log(9) - 5\log(9) = x\log(12)

Step 4: Isolate the Variable $x$

We can isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation.

x\log(9) - x\log(12) = 5\log(9)

Step 5: Factor Out the Common Term

We can factor out the common term $x$ from the left side of the equation.

x(\log(9) - \log(12)) = 5\log(9)

Step 6: Solve for $x$

We can now solve for $x$ by dividing both sides of the equation by the coefficient of $x$.

x = \frac{5\log(9)}{\log(9) - \log(12)}

Step 7: Simplify the Expression

We can simplify the expression by using the properties of logarithms.

x = \frac{5\log(9)}{\log\left(\frac{9}{12}\right)}

Step 8: Evaluate the Expression

We can evaluate the expression by using a calculator or by simplifying it further.

x = \frac{5\log(9)}{\log\left(\frac{1}{4/3}\right)}

Step 9: Simplify the Fraction

We can simplify the fraction by multiplying the numerator and denominator by the reciprocal of the fraction.

x = \frac{5\log(9)}{\log\left(\frac{3}{4}\right)}

Step 10: Use a Calculator to Evaluate the Expression

We can use a calculator to evaluate the expression and find the value of $x$.

x \approx \frac{5(0.9542)}{-0.2877}

Step 11: Simplify the Expression

We can simplify the expression by multiplying the numerator and denominator.

x \approx \frac{4.771}{-0.2877}

Step 12: Evaluate the Expression

We can evaluate the expression by using a calculator.

x \approx -16.6

The final answer is $\boxed{-16.6}$.

Introduction

Solving exponential equations can be a challenging task, especially when dealing with different bases. In our previous article, we solved the equation $9^{x-5}=12^x$ and found the value of $x$. In this article, we will answer some frequently asked questions related to solving exponential equations.

Q: What is the first step in solving an exponential equation?

A: The first step in solving an exponential equation is to take the logarithm of both sides. This will allow us to use the properties of logarithms to simplify the equation.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log_b(M^r) = r\log_b(M)$. This rule allows us to simplify the equation by distributing the logarithm.

Q: How do I isolate the variable $x$ in an exponential equation?

A: To isolate the variable $x$, you need to move all the terms involving $x$ to one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the difference between the natural logarithm and the common logarithm?

A: The natural logarithm (ln) and the common logarithm (log) are both used to take the logarithm of a number. However, the natural logarithm is based on the number $e$, while the common logarithm is based on the number 10.

Q: How do I simplify an expression involving logarithms?

A: To simplify an expression involving logarithms, you can use the properties of logarithms, such as the power rule and the product rule. You can also use a calculator to evaluate the expression.

Q: What is the significance of the base of an exponential equation?

A: The base of an exponential equation is the number that is raised to a power. The base can be any positive number, but it is usually a whole number or a fraction.

Q: How do I solve an exponential equation with a negative exponent?

A: To solve an exponential equation with a negative exponent, you need to rewrite the equation with a positive exponent. You can do this by taking the reciprocal of the base and changing the sign of the exponent.

Q: What is the difference between an exponential equation and a logarithmic equation?

A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. Exponential equations involve a base raised to a power, while logarithmic equations involve a logarithm of a number.

Q: How do I use a calculator to solve an exponential equation?

A: To use a calculator to solve an exponential equation, you need to enter the equation into the calculator and press the "solve" button. You can also use the calculator to evaluate the expression and find the value of the variable.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not taking the logarithm of both sides of the equation
• Not using the properties of logarithms to simplify the equation
• Not isolating the variable $x$ correctly
• Not using a calculator to evaluate the expression

By avoiding these common mistakes, you can ensure that you are solving exponential equations correctly and accurately.

Conclusion

Solving exponential equations can be a challenging task, but with the right techniques and tools, you can solve them accurately and efficiently. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving exponential equations and apply this skill to a wide range of mathematical problems.