Solve The Following Equations:(13) ( 1 3 ) X = 9 \left(\frac{1}{3}\right)^x = 9 ( 3 1 ​ ) X = 9 (14) ( 1 6 ) X = 1 \left(\frac{1}{6}\right)^x = 1 ( 6 1 ​ ) X = 1 (17) ( 1 3 ) X = 1 27 \left(\frac{1}{3}\right)^x = \frac{1}{27} ( 3 1 ​ ) X = 27 1 ​ (18) 3 ( 1 3 ) X = 81 3\left(\frac{1}{3}\right)^x = 81 3 ( 3 1 ​ ) X = 81

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will explore four different exponential equations and provide step-by-step solutions to each of them.

Equation (13): Solving for x in $\left(\frac{1}{3}\right)^x = 9$

To solve this equation, we need to isolate the variable x. We can start by rewriting the equation as:

$$\left(\frac{1}{3}\right)^x = 9$$

We can rewrite 9 as a power of 3, since 9 = 3^2. This gives us:

$$\left(\frac{1}{3}\right)^x = 3^2$$

Now, we can use the property of exponents that states:

$$\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$$

Applying this property to our equation, we get:

$$\frac{1^x}{3^x} = 3^2$$

Simplifying the left-hand side, we get:

$$\frac{1}{3^x} = 3^2$$

Now, we can take the reciprocal of both sides to get:

$$3^x = \frac{1}{3^2}$$

Simplifying the right-hand side, we get:

$$3^x = \frac{1}{9}$$

Now, we can take the logarithm of both sides to solve for x. We can use the logarithm base 3, since we are dealing with powers of 3. This gives us:

$$\log_3(3^x) = \log_3\left(\frac{1}{9}\right)$$

Using the property of logarithms that states:

$$\log_a(a^x) = x$$

We can simplify the left-hand side to get:

$$x = \log_3\left(\frac{1}{9}\right)$$

Now, we can use the change of base formula to rewrite the logarithm in terms of common logarithms:

$$x = \frac{\log\left(\frac{1}{9}\right)}{\log(3)}$$

Simplifying the numerator, we get:

$$x = \frac{\log(1) - \log(9)}{\log(3)}$$

Using the property of logarithms that states:

$$\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$$

We can simplify the numerator to get:

$$x = \frac{\log\left(\frac{1}{9}\right)}{\log(3)}$$

Now, we can use the property of logarithms that states:

$$\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$$

We can simplify the numerator to get:

$$x = \frac{\log(1) - \log(9)}{\log(3)}$$

Simplifying the numerator, we get:

$$x = \frac{-2\log(3)}{\log(3)}$$

Now, we can cancel out the common factor of log(3) to get:

$$x = -2$$

Therefore, the solution to equation (13) is x = -2.

Equation (14): Solving for x in $\left(\frac{1}{6}\right)^x = 1$

To solve this equation, we need to isolate the variable x. We can start by rewriting the equation as:

$$\left(\frac{1}{6}\right)^x = 1$$

We can rewrite 1 as a power of 6, since 1 = 6^0. This gives us:

$$\left(\frac{1}{6}\right)^x = 6^0$$

Now, we can use the property of exponents that states:

$$\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$$

Applying this property to our equation, we get:

$$\frac{1^x}{6^x} = 6^0$$

Simplifying the left-hand side, we get:

$$\frac{1}{6^x} = 1$$

Now, we can take the reciprocal of both sides to get:

$$6^x = 1$$

Now, we can take the logarithm of both sides to solve for x. We can use the logarithm base 6, since we are dealing with powers of 6. This gives us:

$$\log_6(6^x) = \log_6(1)$$

Using the property of logarithms that states:

$$\log_a(a^x) = x$$

We can simplify the left-hand side to get:

$$x = \log_6(1)$$

Now, we can use the property of logarithms that states:

$$\log_a(1) = 0$$

We can simplify the right-hand side to get:

$$x = 0$$

Therefore, the solution to equation (14) is x = 0.

Equation (17): Solving for x in $\left(\frac{1}{3}\right)^x = \frac{1}{27}$

To solve this equation, we need to isolate the variable x. We can start by rewriting the equation as:

$$\left(\frac{1}{3}\right)^x = \frac{1}{27}$$

We can rewrite 1/27 as a power of 3, since 1/27 = 3^(-3). This gives us:

$$\left(\frac{1}{3}\right)^x = 3^{-3}$$

Now, we can use the property of exponents that states:

$$\left(\frac{a}{b}\right)^x = \frac{a^x}{b^x}$$

Applying this property to our equation, we get:

$$\frac{1^x}{3^x} = 3^{-3}$$

Simplifying the left-hand side, we get:

$$\frac{1}{3^x} = 3^{-3}$$

Now, we can take the reciprocal of both sides to get:

$$3^x = 3^3$$

Now, we can take the logarithm of both sides to solve for x. We can use the logarithm base 3, since we are dealing with powers of 3. This gives us:

$$\log_3(3^x) = \log_3(3^3)$$

Using the property of logarithms that states:

$$\log_a(a^x) = x$$

We can simplify the left-hand side to get:

$$x = \log_3(3^3)$$

Now, we can simplify the right-hand side to get:

$$x = 3$$

Therefore, the solution to equation (17) is x = 3.

Equation (18): Solving for x in $3\left(\frac{1}{3}\right)^x = 81$

To solve this equation, we need to isolate the variable x. We can start by rewriting the equation as:

$$3\left(\frac{1}{3}\right)^x = 81$$

We can rewrite 81 as a power of 3, since 81 = 3^4. This gives us:

$$3\left(\frac{1}{3}\right)^x = 3^4$$

Now, we can divide both sides by 3 to get:

$$\left(\frac{1}{3}\right)^x = 3^3$$

Now, we can take the logarithm of both sides to solve for x. We can use the logarithm base 3, since we are dealing with powers of 3. This gives us:

$$\log_3\left(\left(\frac{1}{3}\right)^x\right) = \log_3(3^3)$$

Using the property of logarithms that states:

$$\log_a(a^x) = x$$

We can simplify the left-hand side to get:

$$x = \log_3(3^3)$$

Now, we can simplify the right-hand side to get:

$$x = 3$$

Therefore, the solution to equation (18) is x = 3.

In our previous article, we explored four different exponential equations and provided step-by-step solutions to each of them. In this article, we will answer some common questions that students often have when it comes to solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is an expression of the form a^x, where a is a positive number and x is a variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use various techniques, including rewriting the equation in a more manageable form, using the property of exponents, and taking the logarithm of both sides.

Q: What is the property of exponents?

A: The property of exponents states that:

• a^(m+n) = a^m * a^n
• (a^m)n = a^(m*n)
• a^(-m) = 1/a^m

Q: How do I use the property of exponents to solve an exponential equation?

A: To use the property of exponents to solve an exponential equation, you can rewrite the equation in a more manageable form by using the property of exponents. For example, if you have the equation a^x = b, you can rewrite it as a^(x+1) = ab.

Q: What is the logarithm of an exponential expression?

A: The logarithm of an exponential expression is the exponent to which the base must be raised to produce the given value. For example, if we have the equation 2^x = 8, the logarithm of 8 with base 2 is 3, because 2^3 = 8.

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

A: To use logarithms to solve an exponential equation, you can take the logarithm of both sides of the equation. For example, if you have the equation 2^x = 8, you can take the logarithm of both sides to get:

log(2^x) = log(8)

Using the property of logarithms that states log(a^b) = b*log(a), we can simplify the left-hand side to get:

x*log(2) = log(8)

Now, we can divide both sides by log(2) to get:

x = log(8)/log(2)

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

A: A logarithmic equation is an equation that involves a logarithmic expression, which is an expression of the form log(a), where a is a positive number. An exponential equation, on the other hand, is an equation that involves an exponential expression, which is an expression of the form a^x, where a is a positive number and x is a variable.

Q: Can I use logarithms to solve any exponential equation?

A: Yes, you can use logarithms to solve any exponential equation. However, you need to make sure that the base of the logarithm is the same as the base of the exponential expression.

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

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

• Not rewriting the equation in a more manageable form
• Not using the property of exponents
• Not taking the logarithm of both sides of the equation
• Not using the correct base for the logarithm

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

Conclusion

Solving exponential equations can be a challenging task, but with the right techniques and strategies, you can master it. By using the property of exponents, taking the logarithm of both sides of the equation, and avoiding common mistakes, you can solve exponential equations with ease. Remember to always rewrite the equation in a more manageable form, use the property of exponents, and take the logarithm of both sides of the equation to ensure that you are solving the equation correctly.