Question

Solve for \(x\) to three significant digits. $$e^{5 x}=125$$

Short answer

After simplification and calculation, \(x \approx 0.966\) to three significant digits.

Step-by-step solution

Apply the natural logarithm to both sides of the equation

The equation is of the form \(e^{5x} = 125\). To solve for \(x\), take the natural logarithm (ln) of both sides. This gives us \(\ln(e^{5x}) = \ln(125)\).

Simplify using the property of logarithms

Applying the property of logarithms that \(\ln(e^y) = y\), we simplify the left side to get \(5x = \ln(125)\).

Calculate the natural logarithm of 125

Using a calculator, find the value of \(\ln(125)\), which is approximately 4.828.

Divide both sides of the equation by 5

To isolate \(x\), divide both sides of the equation by 5: \(x = \frac{\ln(125)}{5}\).

Calculate the value of \(x\)

Perform the division to find the value of \(x\): \(x \approx \frac{4.828}{5} = 0.966\).

Key concepts

Natural Logarithm

The natural logarithm, represented as ‘ln’, is a mathematical function that is used to undo the effect of the exponential function, particularly when the base is Euler’s number, denoted as ‘e’. It’s one of the most common logarithmic functions and plays a crucial role in solving equations involving exponents.

For instance, if you have an expression like \(e^y\), taking the natural logarithm of both sides transforms it into \(ln(e^y) = y\), because the natural logarithm and the exponential function are inverses of each other. In the context of our exercise, by applying the natural logarithm to both sides of the equation \(e^{5x} = 125\), we make use of this inverse relationship to extract the exponent from its position and bring it down to a solvable algebraic expression. It’s this powerful property that often makes \(ln\) the method of choice when dealing with exponential equations.

Properties of Logarithms

Logarithms come with a set of properties that simplify complex algebraic manipulations. Understanding these properties can greatly aid in solving equations involving logarithmic terms. The properties that are particularly useful include:

• The Power Rule: \( \text{ln}(x^a) = a \text{ln}(x) \), which lets us move the exponent of the argument to the front as a multiplier.
• The Product Rule: \( \text{ln}(xy) = \text{ln}(x) + \text{ln}(y) \), which converts the log of a product into a sum of logs.
• The Quotient Rule: \( \text{ln}(x/y) = \text{ln}(x) - \text{ln}(y) \), which conversely turns the log of a quotient into a difference of logs.

During the solution to our problem, we apply the Power Rule to deal with the term \( e^{5x} \). By bringing down the exponent in this fashion, we transform it into a simple equation of \( 5x = \text{ln}(125) \). This eliminates the exponent and allows us to isolate and solve for the variable ‘x’.

Exponential Functions

Exponential functions represent continuous growth or decay processes and have the general form \( f(x) = a^x \), where ‘a’ is a constant base. In our exercise, the base is the special mathematical constant ‘e’, which is approximately equal to 2.71828. This constant is particularly convenient when dealing with continuous growth rates or compound interest calculations.

An important aspect of exponential functions is their one-to-one nature, which makes them invertible—hence, allowing us to use logarithms to solve for their exponents. Moreover, exponential functions grow much more rapidly than polynomial functions. This is why, even though the numbers involved may not seem large, you can still end up with significant quantities when dealing with exponential equations.

To find a specific growth factor or the time at which a certain level is reached, you often need to solve the exponential equation for the exponent ‘x’. In our example, finding the value of ‘x’ required transforming the exponential equation into a more manageable form using the natural logarithm.