Solve For { X $}$ Using Logarithms To The Fourth Decimal Place.${ 5^{x+7} = 3 }${$ X = \square $}$
**Solving Exponential Equations with Logarithms: A Step-by-Step Guide** ===========================================================
Introduction
Exponential equations can be challenging to solve, especially when they involve large or complex numbers. However, with the help of logarithms, we can simplify these equations and find their solutions. In this article, we will explore how to solve exponential equations using logarithms, with a focus on solving for x in the equation 5^(x+7) = 3.
What are Logarithms?
Before we dive into solving exponential equations, let's briefly review what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have an equation in the form a^x = b, we can take the logarithm of both sides to solve for x. There are two main types of logarithms: common logarithms (base 10) and natural logarithms (base e).
Solving Exponential Equations with Logarithms
Now that we have a basic understanding of logarithms, let's apply this knowledge to solve the equation 5^(x+7) = 3.
Step 1: Take the Logarithm of Both Sides
To solve for x, we need to get rid of the exponent. We can do this by taking the logarithm of both sides of the equation. Let's use the natural logarithm (base e) for this example.
\ln(5^{x+7}) = \ln(3)
Step 2: Apply the Power Rule of Logarithms
The power rule of logarithms states that \ln(a^b) = b \ln(a). We can apply this rule to the left side of the equation.
(x+7) \ln(5) = \ln(3)
Step 3: Distribute the Natural Logarithm
Now that we have the power rule applied, we can distribute the natural logarithm to the terms inside the parentheses.
x \ln(5) + 7 \ln(5) = \ln(3)
Step 4: Isolate x
To isolate x, we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 7 \ln(5) from both sides.
x \ln(5) = \ln(3) - 7 \ln(5)
Step 5: Solve for x
Finally, we can solve for x by dividing both sides of the equation by \ln(5).
x = \frac{\ln(3) - 7 \ln(5)}{\ln(5)}
Example Solution
Let's plug in some values to see how this works in practice. Suppose we want to find the value of x in the equation 5^(x+7) = 3.
\ln(5^{x+7}) = \ln(3)
(x+7) \ln(5) = \ln(3)
x \ln(5) + 7 \ln(5) = \ln(3)
x \ln(5) = \ln(3) - 7 \ln(5)
x = \frac{\ln(3) - 7 \ln(5)}{\ln(5)}
Using a calculator, we can find the value of x to be approximately -1.6094.
Conclusion
Solving exponential equations with logarithms can be a powerful tool for finding solutions to complex equations. By following the steps outlined in this article, we can simplify these equations and find their solutions. Remember to take the logarithm of both sides, apply the power rule, distribute the natural logarithm, isolate x, and finally solve for x.
Frequently Asked Questions
Q: What is the difference between common logarithms and natural logarithms?
A: Common logarithms have a base of 10, while natural logarithms have a base of e.
Q: Can I use any type of logarithm to solve exponential equations?
A: No, you should use the logarithm that is the inverse of the base of the exponent. For example, if the equation is in the form a^x = b, you should use the logarithm with base a.
Q: How do I know which type of logarithm to use?
A: The type of logarithm you use will depend on the base of the exponent. If the base is a common number (such as 10), you may want to use common logarithms. If the base is a more complex number (such as e), you may want to use natural logarithms.
Q: Can I use logarithms to solve equations with negative exponents?
A: Yes, you can use logarithms to solve equations with negative exponents. However, you will need to use the property of logarithms that states \ln(a^(-b)) = -b \ln(a).
Q: How do I apply the power rule of logarithms?
A: To apply the power rule of logarithms, you need to multiply the exponent by the logarithm of the base. For example, if you have the equation \ln(a^b) = c, you can apply the power rule by multiplying b by \ln(a) to get b \ln(a) = c.