Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 8^{-x}=1.2 $$

Short answer

The exact form of the solution is \(x = - \frac{\text{ln}(1.2)}{\text{ln}(8)}\).

Step-by-step solution

Rewrite the exponential equation

Rewrite the given equation as \[8^{-x} = 1.2\].

Take the natural logarithm of both sides

Apply the natural logarithm (ln) to both sides of the equation to get \[\text{ln}(8^{-x}) = \text{ln}(1.2)\].

Use the properties of logarithms

Use the property of logarithms that states \(\text{ln}(a^b) = b \text{ln}(a)\). This simplifies the equation to \[-x \text{ln}(8) = \text{ln}(1.2)\].

Solve for \(x\)

Divide both sides by \(-\text{ln}(8)\) to isolate \(x\) and solve the equation: \[x = - \frac{\text{ln}(1.2)}{\text{ln}(8)}\].

Key concepts

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}(x)\), is a logarithm to the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. Natural logarithms are widely used in mathematics, especially in solving exponential equations. To understand how \(\text{ln}(x)\) works, we need to know that if \(\text{ln}(x) = y\), then \(e^y = x\). This unique property of \(e\) makes natural logarithms particularly useful in calculus and higher-level math.

Logarithmic Properties

Logarithms have several important properties that make solving equations much easier. One key property is the power rule, which states that \(\text{ln}(a^b) = b \text{ln}(a)\). This means if you take the logarithm of a number raised to a power, you can bring the exponent in front as a multiplier. Another useful property is the product rule, \(\text{ln}(ab) = \text{ln}(a) + \text{ln}(b)\), and the quotient rule, \(\text{ln}\frac{a}{b} = \text{ln}(a) - \text{ln}(b)\). These properties help break down complex logarithmic expressions into simpler parts, making it easier to solve for the unknown variable.

Irrational Solutions

An irrational solution is a solution that cannot be expressed as a simple fraction. In the context of logarithmic and exponential equations, irrational solutions often come in 'exact form.' This means you don't approximate the solution as a decimal. For instance, if you solve an exponential equation and your answer involves \(\text{ln}(8)\) or \(\text{ln}(1.2)\), you leave it as is—without converting it to a numerical value. Keeping answers in exact form is crucial for maintaining accuracy in mathematical problems. Therefore, an exact form involving natural logs or other irrational numbers is standard in many math problems.

Exact Form

In mathematical solutions, 'exact form' refers to expressing the answer in a way that maintains all original values and constants, such as \(\text{ln}(8)\) or \(\text{ln}(1.2)\). For example, if you solve \(8^{-x} = 1.2\) and find \(x = - \frac{\text{ln}(1.2)}{\text{ln}(8)}\), you keep it in this exact form. This practice ensures the highest possible precision, avoiding the pitfalls of rounding errors that can occur with decimal approximations. Always aim to present your solutions without unnecessary approximations for accurate and reliable results.