Question

In Exercises, solve for \(x\) or \(t\). $$ 2 \ln 4 x=7 $$

Short answer

The solution to the equation is \(x = \sqrt{e^7 / 16}\).

Step-by-step solution

Rewrite the Equation

The given equation is \(2 \ln(4x) = 7\). Firstly, the left side of the equation can be simplified using properties of logarithm. Since the coefficient of a logarithm can be rewritten as an exponent of the argument, we rewrite the equation as \(\ln[(4x)^2] = 7\).

Simplify the Equation

Now simplify the equation. The square of \(4x\) is \(16x^2\), so the equation becomes \(\ln(16x^2) = 7\).

Convert to Exponential Form

Next, convert the logarithmic equation to an exponential one using the fact that \(\ln a = b\) is the same as \(e^b = a\), where \(e\) is the base of the natural logarithm. By doing this, we get \(e^7 = 16x^2\).

Solve for x

Finally, isolate \(x\) by dividing both sides of the equation by 16 and taking the square root: \(x = \sqrt{e^7 / 16}\).

Key concepts

Natural Logarithm

The natural logarithm, often denoted by "ln," is a special type of logarithm where the base is the mathematical constant \( e \), approximately equal to 2.71828. Unlike other logarithms that might use a different base, such as 10 for the common logarithm, the natural logarithm is particularly useful in calculus and natural sciences.
Its defining property is that it reverses the action of exponentiation with base \( e \).

• If \( \ln(a) = b \), it means that \( e^b = a \).
• This feature is critical for simplifying complex equations and solving for unknowns.

In the context of the exercise, understanding the natural logarithm is essential for converting the logarithmic equation into an exponential form, which is a pivotal step in finding the solution.

Exponential Form

Turning a logarithmic equation into an exponential form is a helpful technique for solving it. It allows you to "switch" from a log-based view to exponential growth.
This process is based on the relationship: if \( \ln(a) = b \), then \( e^b = a \).

• This signifies an inversion of the logarithmic operation, which is key to solving the problem.
• By converting \( \ln(16x^2) = 7 \) to the exponential form \( e^7 = 16x^2 \), you can then isolate and solve for \( x \).

This step showcases the flexibility of logarithmic expressions, allowing for easier manipulation of equations to ultimately find solutions.

Properties of Logarithms

To solve logarithmic equations effectively, grasp the properties of logarithms. These rules simplify expressions and equations involving logarithms. For instance, the power rule is extremely useful.
The power rule states that \( c \ln(a) = \ln(a^c) \), allowing the coefficient in front of a logarithmic expression to become an exponent.

• In the exercise, this property helps rewrite \( 2 \ln(4x) \) as \( \ln((4x)^2) \).
• This reduction simplifies manipulation of the terms.

Recognizing these properties allows you to streamline calculations and convert complex equations into simpler forms, enhancing problem-solving efficiency.

Solving Equations

Solving equations with logarithmic expressions may seem daunting, but following a step-by-step method simplifies the process. After rewriting and simplifying the equation, the main task is to isolate the variable.
In our case, start by converting the logarithmic equation to its exponential form as shown. Then, to solve for \(x\), further manipulation is required.

• From \( e^7 = 16x^2 \), divide both sides by 16 to get \( e^7/16 = x^2 \).
• To find \( x \), take the square root: \( x = \sqrt{e^7/16} \).

Applying logical steps and using logarithmic properties ensures a comprehensive solution to these types of equations.