Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 5\left(2^{3 x}\right)=8 $$

Short answer

x = \frac{\ln \left(\frac{8}{5}\right)}{3 \cdot \ln(2)}

Step-by-step solution

- Divide both sides by 5

First, isolate the exponential part of the equation. To do this, divide both sides by 5: \[ 5 \left(2^{3x}\right) = 8 \] \[ 2^{3x} = \frac{8}{5} \]

- Apply natural logarithm to both sides

Next, apply the natural logarithm (ln) to both sides of the equation to get rid of the exponent: \[ \ln \left(2^{3x}\right) = \ln \left(\frac{8}{5}\right) \]

- Use logarithm power rule

Apply the power rule of logarithms which states \( \ln(a^b) = b \cdot \ln(a) \) to the left side: \[ 3x \cdot \ln(2) = \ln \left(\frac{8}{5}\right) \]

- Solve for x

Finally, solve for x by dividing both sides by \(3 \cdot \ln(2)\): \[ x = \frac{\ln \left(\frac{8}{5}\right)}{3 \cdot \ln(2)} \]

Key concepts

Natural logarithm

Natural logarithms are logarithms with base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. They are denoted as \(\text{ln}\) and are fundamental in solving exponential equations. By applying the natural logarithm to both sides of an equation, we take advantage of its properties to simplify the expressions, particularly those involving exponents. For instance, in our exercise, \(\text{ln}(2^{3x}) = \text{ln}(\frac{8}{5})\), we use the natural logarithm to manipulate the exponential expression into a more solvable form.

Logarithm power rule

The logarithm power rule is an essential identity in logarithmic mathematics. It states that for any real numbers 'a' and 'b', \(\text{ln}(a^b) = b \text{ln}(a)\). This rule simplifies exponents inside a logarithm by converting them into a multiplier. In the given problem, after applying the natural logarithm, we use the power rule on \(\text{ln}(2^{3x}) \), transforming it into \({3x \text{ln}(2)}\). This step is crucial for isolating the variable 'x', making it easier to solve the equation.

Irrational solutions

In mathematics, an irrational solution is one that cannot be expressed as a simple fraction. These solutions typically involve transcendental numbers like 'e'. For the current problem, the solution for 'x' takes the form of \(\frac{\text{ln} \left(\frac{8}{5}\right)}{3 \text{ln}(2)}\), which is irrational. This exact form is preferred instead of a decimal approximation to preserve the precision of the answer in mathematical contexts. Thus, understanding irrational numbers and their representations is key in higher mathematics.

Solving exponential equations

To solve exponential equations like the provided example, follow a systematic approach:
\(\text{1. Isolate the exponential expression: Divide both sides by 5, yielding } 2^{3x} = \frac{8}{5}\).
2. Apply the natural logarithm to both sides: Using \(\text{ln}(2^{3x}) = \text{ln}(\frac{8}{5})\).
3. Utilize the logarithm power rule: Transform \(\text{ln}(2^{3x})\) to \({3x \text{ln}(2)}\).
4. Solve for the variable: Finally, isolate x by dividing, resulting in \({x = \frac{\text{ln}\text{(} \frac{8}{5} \text{)}}{3 \text{ln}(2)}}\).
Each of these steps brings you closer to finding the exact solution while understanding the underlying concepts.