145. Without using a calculator, determine which is the greater number: log₄ 60 or log₃ 40.

Verified step by step guidance

1. Recall that \( \log_a b \) represents the logarithm of \( b \) with base \( a \), which answers the question: "To what power must we raise \( a \) to get \( b \)?".
2. To compare \( \log_4 60 \) and \( \log_3 40 \) without a calculator, express both logarithms in terms of a common base, such as the natural logarithm \( \ln \), using the change of base formula: \( \log_a b = \frac{\ln b}{\ln a} \).
3. Rewrite the expressions as \( \log_4 60 = \frac{\ln 60}{\ln 4} \) and \( \log_3 40 = \frac{\ln 40}{\ln 3} \).
4. Estimate the values of \( \ln 60 \), \( \ln 40 \), \( \ln 4 \), and \( \ln 3 \) by recognizing nearby known values or using properties of logarithms (for example, \( \ln 4 = 2 \ln 2 \), \( \ln 3 \) is known to be about 1.1, etc.).
5. Compare the two fractions \( \frac{\ln 60}{\ln 4} \) and \( \frac{\ln 40}{\ln 3} \) by cross-multiplying or estimating to determine which is greater.

Key Concepts

Logarithm Definition and Properties

A logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this helps compare logarithms by interpreting their values relative to their bases and arguments.

Change of Base Formula

The change of base formula, log_b(a) = log_c(a) / log_c(b), allows rewriting logarithms with different bases into a common base, facilitating direct comparison without a calculator.

Estimating Logarithmic Values

Estimating logarithms involves approximating values using known logs or inequalities, such as recognizing that log_b(a) increases as a increases and decreases as b increases, enabling comparison without exact calculation.