Question

Solve each equation. $$ e^{3 x}=10 $$

Short answer

x = \frac{\ln(10)}{3}

Step-by-step solution

- Understand the given equation

The equation provided is an exponential equation: \[ e^{3x} = 10 \] where the base is the constant \( e \), and the exponent is \( 3x \).

- Apply the natural logarithm

To solve for \( x \), apply the natural logarithm (ln) to both sides of the equation. This will help in bringing the exponent down. \[ \ln(e^{3x}) = \ln(10) \] When you apply the natural logarithm to \( e \) raised to a power, you can use the property \( \ln(e^y) = y \).

- Simplify using logarithm properties

Using the property that \( \ln(e^y) = y \), simplify the left side of the equation: \[ 3x = \ln(10) \].

- Solve for \( x \)

To isolate \( x \), divide both sides of the equation by 3: \[ x = \frac{\ln(10)}{3} \].

Key concepts

exponential functions

Exponential functions are a type of mathematical function where the variable is in the exponent. For example, in the function \(f(x) = e^{3x}\), the variable \(x\) is in the exponent position.
Exponential functions are important in many fields, such as biology for modeling population growth, finance for calculating compound interest, and physics for describing radioactive decay.
One of the most common bases for exponential functions is Euler's number, \(e\), which is approximately equal to 2.71828. This specific base often appears in natural logarithms and in various natural processes.

natural logarithm

The natural logarithm, denoted as \(\ln\), is the inverse function of the natural exponential function. When you see \(\ln(x)\), it is asking the question: 'To what power must \(e\) be raised, to get \(x\)?'
For example, \(\ln(e) = 1\) because \(e^1 = e\). Therefore, natural logarithms are very useful when working with exponential equations, especially those involving the constant \(e\).
In our problem, applying \(\ln\) to both sides of the equation allows us to pull down the exponent for easier manipulation. This brings down the exponential equation \(e^{3x} = 10\) to \(3x = \ln(10)\).

logarithmic properties

Logarithmic properties are essential tools for simplifying and solving logarithmic and exponential equations. Here are some key logarithmic properties:

• \(\ln(e^y) = y\): This property shows that applying the natural logarithm to \(e\) raised to any power will return that power.
• \(\ln(ab) = \ln(a) + \ln(b)\): This property helps when we are dealing with the product of two values inside a logarithm.
• \(\ln(a/b) = \ln(a) - \ln(b)\): This property is useful when dealing with the division of two values inside a logarithm.
• \(\ln(a^b) = b\ln(a)\): This property allows us to bring an exponent out in front of a logarithm.

In our example, we used \(\ln(e^{3x}) = 3x\) to simplify the equation during the solving process.

equation solving steps

Solving exponential equations often requires a series of logical steps to isolate the variable. Here is how we solved the given equation step-by-step:

• **Step 1:** Understand the given equation. Recognize that \(e^{3x} = 10\) is an exponential equation with a base of \(e\) and an exponent of \(3x\).
• **Step 2:** Apply the natural logarithm to both sides of the equation. This helps to bring the exponent down, resulting in \(\ln(e^{3x}) = \ln(10)\).
• **Step 3:** Simplify using logarithm properties. Using the property \(\ln(e^y) = y\), we simplify the left side to get \(3x = \ln(10)\).
• **Step 4:** Solve for \(x\). Isolate \(x\) by dividing both sides by 3. Thus, \(x = \frac{\ln(10)}{3}\).

Remember, solving exponential equations often involves using logarithmic identities to simplify and isolate the variable.