Question

Evaluate the expression without using a calculator.\(\log _{8} 2\)

Short answer

The value of \(\log_8 2\) is \(1/3\).

Step-by-step solution

Convert the Logarithm to Exponential Form

Recall that a logarithm \(\log_b a = c\) can be rewritten in exponential form as \(b^c = a\). Let's apply this to the equation \(\log_8 2\). So, if \(\log _{8} 2 = x\), we can rewrite this as \(8^x = 2\).

Express 8 and 2 with the same base

We can express both 8 and 2 as powers of 2. Remember 8 can be written as \(2^3\) and 2 can be written as \(2^1\). So, we can rewrite our equation as \((2^3)^x = 2^1\).

Solve for x

Simplify \((2^3)^x = 2^1\) to \(2^{3x} = 2^1\). In an equation of the form \(a^m = a^n\), one can say that \(m = n\). Therefore, \(3x = 1\). Solving for \(x\), we get \(x = 1/3\).

Key concepts

Exponential Functions

Exponential functions are a fundamental part of mathematics, especially when dealing with growth patterns and interest calculations. An exponential function is typically in the form of \(b^x\), where \(b\) is a positive real number, known as the base, and \(x\) is any real number. The beauty of exponential functions is in how they grow—quickly and unpredictably.

• Exponential growth: Occurs when the base \(b > 1\). If something grows exponentially, it increases rapidly at a rate proportional to its current value.
• Exponential decay: Happens when the base \(0 < b < 1\). In this case, the value decreases rapidly.

Exponential functions are pivotal in converting logarithmic expressions into a form that can be more easily manipulated and understood. For example, converting \(\log_8 2\) into \(8^x = 2\) allows us to clearly see the underlying relationships between the numbers.

Logarithmic Properties

Logarithms are the inverse operations of exponentials. They help to "undo" an exponent. The equation \(\log_b a = c\) tells us the power \(c\) to which the base \(b\) must be raised to produce the value \(a\). Some key properties of logarithms are very useful in simplifying complex mathematical problems.

• Product Property: \(\log_b (mn) = \log_b m + \log_b n\). Useful for combining logs.
• Quotient Property: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\). Handy for splitting logs.
• Power Property: \(\log_b (m^n) = n \cdot \log_b m\). Breaks exponents into multipliers.

By using the exponential form of a logarithm, as seen with the conversion in \(\log_8 2 = x\) to \(8^x = 2\), we can utilize these properties to make calculations more straightforward and achieve solutions more efficiently.

Algebraic Manipulation

Algebraic manipulation involves the skillful transformation and rearrangement of equations in order to find unknowns or to simplify expressions. In the context of logarithmic and exponential equations, this often involves changing the form of the expression to reveal hidden pathways to the solution.
In the problem \(\log_8 2 = x\), we express both terms \(8\) and \(2\) using the same base, which is 2. Recognizing that \(8 = 2^3\) allows us to rewrite \((2^3)^x = 2^1\), streamlining the problem into a simpler equivalent form, \(2^{3x} = 2^1\).

• Common principles include the use of equivalent expressions to simplify equations.
• Match bases when dealing with exponential terms to directly equate the exponents.
• Solving for unknown variables becomes an easier task once the expressions are simplified.

Algebraic manipulation is thus fundamental in breaking down complex mathematical problems into understandable and solvable parts.