Exponential & Logarithmic Equations: Videos & Practice Problems

To solve exponential equations, rewrite both sides to have the same base, allowing you to set the exponents equal. For example, if $\mathrm{16}=2^x$, rewrite $\mathrm{16}$ as $2^4$ to find $x=4$. If bases differ, use logarithms to isolate the variable. For logarithmic equations, set equal logs of the same base or convert single logs to exponential form for solving.

Solving Exponential Equations Using Like Bases

When working with exponential functions, evaluating the function for a specific value of \( x \) is straightforward. However, when the function is set equal to a value, the goal shifts to finding the value of \( x \) that satisfies the equation. This process involves rewriting both sides of the equation to have the same base, allowing us to solve it as a basic linear equation.

For example, consider the equation \( 16 = 2^x \). To solve for \( x \), we first rewrite \( 16 \) as a power of \( 2 \). Since \( 16 = 2^4 \), we can rewrite the equation as \( 2^4 = 2^x \). With both sides having the same base, we can set the exponents equal to each other, resulting in \( 4 = x \). Thus, the solution is \( x = 4 \).

In another example, we have \( 64 = 2^x \). To express \( 64 \) as a power of \( 2 \), we can break it down: \( 64 = 8^2 \) and \( 8 = 2^3 \). Therefore, \( 64 = (2^3)^2 = 2^6 \). Now, we have \( 2^6 = 2^x \), leading to \( 6 = x \) and the solution \( x = 6 \).

Next, consider the equation \( 5^{x+1} = \sqrt{5} \). Here, we recognize that \( \sqrt{5} \) can be rewritten as \( 5^{1/2} \). This gives us \( 5^{x+1} = 5^{1/2} \). Setting the exponents equal results in the equation \( x + 1 = \frac{1}{2} \). Solving for \( x \) by subtracting \( 1 \) from both sides yields \( x = \frac{1}{2} - 1 = -\frac{1}{2} \).

Lastly, consider \( 27 = 9^x \). Since \( 9 \) can be expressed as \( 3^2 \), we rewrite the equation as \( 27 = (3^2)^x = 3^{2x} \). Knowing that \( 27 = 3^3 \), we can set the bases equal: \( 3^3 = 3^{2x} \). This leads to the equation \( 3 = 2x \). Dividing both sides by \( 2 \) gives \( x = \frac{3}{2} \).

In summary, solving exponential equations often involves rewriting the expressions to have the same base, allowing for straightforward comparison of exponents. This method simplifies the process and enables the use of basic algebraic techniques to find the value of \( x \).

Solve the exponential equation.
$4^{x+7}=16$

• A. $2$
• B. $5$
• C. $9$
• D. $-5$

Answer: (D)

Solve the exponential equation.
$100^{x}=10^{x+17}$

• A. $17$
• B. $34$
• C. $8.5$
• D. $0$

Answer: (A)

Solve the exponential equation.
$81^{x+1}=27^{x+5}$

• A. $-13$
• B. $11$
• C. $14$
• D. $3$

Answer: (B)

Solving Exponential Equations Using Logs

Exponential equations can often be solved by rewriting both sides to have the same base, allowing us to set the exponents equal to each other. However, when faced with an equation like \( 17 = 2^x \), where it’s not straightforward to express 17 as a power of 2, logarithms provide a useful alternative. By applying logarithmic properties, we can solve for \( x \) without needing to find complex decimal values.

To illustrate this process, consider the equation \( 10^{x + 64} = 100 \). The first step is to isolate the exponential expression. By subtracting 64 from both sides, we simplify the equation to \( 10^x = 36 \). Next, we determine which logarithm to use. Since the base of the exponential expression is 10, we apply the common logarithm (log base 10) to both sides, resulting in \( \log(10^x) = \log(36) \).

Utilizing the power rule of logarithms, we can bring the exponent \( x \) in front, yielding \( x \cdot \log(10) = \log(36) \). Recognizing that \( \log(10) = 1 \), we simplify this to \( x = \log(36) \). This expression is valid as a final answer, but if an approximation is required, we can calculate \( \log(36) \) using a calculator, which gives approximately \( x \approx 1.56 \).

In another example, consider the equation \( 3 = 2^{x + 1} \). Here, the exponential expression is already isolated, so we proceed directly to taking the natural logarithm (since the base is not 10). This gives us \( \ln(3) = \ln(2^{x + 1}) \). Again, applying the power rule allows us to rewrite this as \( \ln(3) = (x + 1) \cdot \ln(2) \).

To isolate \( x \), we divide both sides by \( \ln(2) \), resulting in \( \frac{\ln(3)}{\ln(2)} = x + 1 \). Finally, we subtract 1 from both sides to find \( x = \frac{\ln(3)}{\ln(2)} - 1 \). This expression, while it may appear complex, represents a constant value. If needed, we can approximate this using a calculator, yielding \( x \approx 0.58 \).

By mastering the use of logarithms and understanding how to manipulate exponential equations, we can confidently tackle a wide range of problems involving exponential relationships.

Solve the exponential equation.
$2\cdot10^{3x}=5000$

• A. $x=3.40$
• B. $x=10.19$
• C. $x=0.0001$
• D. $x=1.13$

Answer: (D)

Solve the exponential equation.
$900=10^{x+17}$

• A. $x=-14.05$
• B. $x=2.95$
• C. $x=0.17$
• D. $x=1.72$

Answer: (A)

Solve the exponential equation.
$e^{2x+5}=8$

• A. $x=-1.46$
• B. $x=-1.11$
• C. $x=-0.22$
• D. $x=1.39$

Answer: (A)

Solve the exponential equation.
$7^{2x^2-8}=1$

• A. $x=0.51$
• B. $x=\pm 2$
• C. $x=\pm 2.83$
• D. $x=2.23$

Answer: (B)

Solving Logarithmic Equations

Logarithmic equations can be simplified into two main types, making them manageable to solve. The first type involves two logarithms of the same base set equal to each other, while the second type consists of a single logarithm set equal to a constant. Both types ultimately reduce to solving basic linear equations.

For the first type, when you have two logarithms with the same base, you can set the arguments of the logarithms equal to each other. For example, if you have log2(x + 1) = log2(5), you can simplify this to x + 1 = 5. Solving this linear equation gives x = 4.

In a more complex scenario, such as ln(x + 4) - ln(2) = ln(8), you can apply the properties of logarithms to condense the equation. Using the quotient rule, this can be rewritten as ln((x + 4)/2) = ln(8). Since both sides have the same base (base e), you can set the arguments equal: (x + 4)/2 = 8. Multiplying both sides by 2 results in x + 4 = 16, and isolating x gives x = 12.

The second type of logarithmic equation involves a single logarithm set equal to a constant, such as log2(4x) = 5. To solve this, first express the logarithm in exponential form. This means rewriting the equation as 25 = 4x. Calculating 25 gives 32, leading to 32 = 4x. Dividing both sides by 4 results in x = 8.

It is crucial to verify your solution by substituting it back into the original logarithmic expression. For instance, substituting x = 8 into 4x gives 4(8) = 32, which is positive. Since logarithms are only defined for positive arguments, this confirms that x = 8 is indeed a valid solution.

In summary, solving logarithmic equations involves recognizing the type of equation you are dealing with, applying logarithmic properties, converting to exponential form when necessary, and ensuring that the solutions yield positive arguments for the logarithms.

Solve the logarithmic equation.
$\log_3\left(3x+9\right)=\log_35+\log_312$

• A. $20$
• B. $17$
• C. $1$
• D. No Solution

Answer: (B)

Solve the logarithmic equation.
$\log\left(x+2\right)+\log2=3$

• A. $498$
• B. $1998$
• C. $6$
• D. No Solution

Answer: (A)

Solve the logarithmic equation.
$\log_7\left(6x+13\right)=2$

• A. $3$
• B. $19.17$
• C. $6$
• D. No Solution

Answer: (C)

Here’s what students ask on this topic:

How do you solve exponential equations by rewriting each side to have the same base?

To solve exponential equations by rewriting each side to have the same base, follow these steps:

1. Identify the base of the exponential expression on one side of the equation.

2. Rewrite the other side of the equation using the same base. For example, if you have 16 = 2^x, rewrite 16 as 2⁴.

3. Once both sides have the same base, set the exponents equal to each other. In this case, 4 = x.

4. Solve the resulting linear equation for the variable.

This method works well when the numbers can be easily rewritten with the same base.

How do you solve exponential equations using logarithms?

To solve exponential equations using logarithms, follow these steps:

1. Isolate the exponential expression on one side of the equation.

2. Take the logarithm of both sides. Use the common log (log) if the base is 10, or the natural log (ln) for other bases.

3. Use the power rule of logarithms to move the exponent in front of the log. For example, if you have 2^x = 17, take the natural log of both sides to get ln(2^x) = ln(17), which simplifies to x ln(2) = ln(17).

4. Solve for the variable by isolating it. In this case, x = ln(17) / ln(2).

5. If needed, use a calculator to approximate the value.

How do you solve logarithmic equations with the same base?

To solve logarithmic equations with the same base, follow these steps:

1. If you have two logs of the same base set equal to each other, set the arguments of the logs equal. For example, if log₂(x + 1) = log₂(5), then x + 1 = 5.

2. Solve the resulting linear equation for the variable. In this case, x = 4.

3. If the equation involves subtraction or addition of logs, use log properties to condense them into a single log before setting the arguments equal.

This method simplifies the equation to a basic linear form, making it easier to solve.

How do you solve logarithmic equations by converting to exponential form?

To solve logarithmic equations by converting to exponential form, follow these steps:

1. Isolate the logarithmic expression on one side of the equation.

2. Convert the log equation to its exponential form. For example, if you have log₂(4x) = 5, rewrite it as 2⁵ = 4x.

3. Simplify the exponential expression. In this case, 2⁵ = 32, so 32 = 4x.

4. Solve the resulting linear equation for the variable. Here, x = 32 / 4, which simplifies to x = 8.

5. Check your solution by substituting it back into the original equation to ensure it is valid.

What are the steps to solve exponential equations with different bases using logarithms?

To solve exponential equations with different bases using logarithms, follow these steps:

1. Isolate the exponential expression on one side of the equation.

2. Take the logarithm of both sides. Use the common log (log) if the base is 10, or the natural log (ln) for other bases.

3. Use the power rule of logarithms to move the exponent in front of the log. For example, if you have 3^x = 7, take the natural log of both sides to get ln(3^x) = ln(7), which simplifies to x ln(3) = ln(7).

4. Solve for the variable by isolating it. In this case, x = ln(7) / ln(3).

5. If needed, use a calculator to approximate the value.