Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 0.3\left(4^{0.2 x}\right)=0.2 $$

Short answer

x = \frac{\text{ln}(2/3)}{0.2 \text{ln}(4)}

Step-by-step solution

Isolate the Exponential Term

Start by dividing both sides of the equation by 0.3 to isolate the exponential term. ewline ewline ewline $$ 4^{0.2x} = \frac{0.2}{0.3} $$ ewline ewline ewline This simplifies to ewline $$ 4^{0.2x} = \frac{2}{3} $$

Apply Logarithm to Both Sides

Take the natural logarithm (ln) of both sides to bring the exponent down: ewline ewline $$ \text{ln}\big(4^{0.2x}\big) = \text{ln}\big(\frac{2}{3}\big) $$

Use Logarithm Power Rule

Apply the power rule of logarithms, \text{ln}(a^b) = b \text{ln}(a), to the left side: ewline ewline $$ 0.2x \text{ln}(4) = \text{ln}\big(\frac{2}{3}\big) $$

Solve for x

Isolate x by dividing both sides by 0.2 \text{ln}(4): ewline $$ x = \frac{\text{ln}\big(\frac{2}{3}\big)}{0.2 \text{ln}(4)} $$ ewline This is the exact form of the solution.

Key concepts

Exact Form Solution

An 'exact form solution' refers to solving an equation without approximating or rounding the answer. In mathematical problems, especially with exponential equations, maintaining an exact form is crucial for accuracy.

For instance, if we solve the given exponential equation and express the solution in exact form, the answer would be written with all terms and operations intact, without converting to decimal approximation. Using logarithms to express the solution helps keep it exact. For example, instead of converting a logarithmic value to a decimal, we leave it as a logarithmic expression.

In our exercise, the exact form solution is represented as: \[ x = \frac{{\text{ln}\big(\frac{2}{3}\big)}}{0.2 \text{ln}(4)} \]. This maintains the precision and is preferred because it leaves no room for rounding errors.

Natural Logarithm

The 'natural logarithm', denoted as \(\text{ln}\), is a logarithm with the base of Euler's number \(e\) (approximately 2.71828). Natural logarithms are especially useful in calculus and solving exponential equations because \(\text{ln}(e^x) = x\).

Natural logarithms have properties that simplify the manipulation of exponential terms. For example, taking the natural logarithm of both sides of an equation helps in bringing down exponents, making the equation easier to solve.

In our exercise, we used \(\text{ln}\) to handle the exponential term \(4^{0.2x}\) by writing: \[ \text{ln}\big(4^{0.2x}\big) = \text{ln}\big(\frac{2}{3}\big) \]. This allowed us to leverage the logarithm power rule effectively.

Logarithm Power Rule

The 'logarithm power rule' is a fundamental property of logarithms which states that \(\text{ln}(a^b) = b \text{ln}(a)\). This rule helps to move the exponent in the argument of the logarithm to the front as a coefficient.

For example, in the exercise, the power rule was applied to transform \(\text{ln}(4^{0.2x})\) to \(0.2x \text{ln}(4)\). This conversion is essential as it simplifies the process of solving for the variable 'x':

• It converts the exponentiation into multiplication, making it easier to isolate the variable.
• It changes a complex equation into a simpler linear form.

Utilizing it in the solution: \[ \text{ln}\big(4^{0.2x}\big) = \text{ln}\big(\frac{2}{3}\big) \] becomes \[ 0.2x \text{ln}(4) = \text{ln}\big(\frac{2}{3}\big) \], thus making ‘x’ easier to solve.

Solving Exponential Equations

Solving exponential equations involves isolating the exponential term and applying logarithms to simplify and solve for the variable. Here’s a step-by-step outline to tackle problems like this:

1. **Isolate the Exponential Term**:
Begin by isolating the term with the exponent. For our equation, divide both sides by 0.3: \(4^{0.2x} = \frac{2}{3}\).

2. **Apply Logarithm**:
Take the natural logarithm on both sides: \(\text{ln}\big(4^{0.2x}\big) = \text{ln}\big(\frac{2}{3}\big)\). This allows us to use logarithmic properties to manage the exponent.

3. **Use the Power Rule**:
Use the power rule to bring the exponent down: \(0.2x \text{ln}(4) = \text{ln}\big(\frac{2}{3}\big)\).

4. **Solve for the Variable**:
Isolate the variable by dividing both sides by the coefficient of 'x': \(\frac{\text{ln}\big(\frac{2}{3}\big)}{0.2 \text{ln}(4)}\). This gives the exact form solution for 'x'.

This method ensures precision and helps avoid approximation errors often encountered with exponential functions.