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

Verified step by step guidance

1. Step 1: Recall the change of base formula for logarithms: ${\log }_{b}a=\frac{{\log }_{k}a}{{\log }_{k}b}$, where $k$ is any positive number.
2. Step 2: Apply the change of base formula to both logarithms using base 10: ${\log }_{4}60=\frac{{\log }_{10}60}{{\log }_{10}4}$ and ${\log }_{3}40=\frac{{\log }_{10}40}{{\log }_{10}3}$.
3. Step 3: Compare the two expressions by considering the numerators and denominators separately. Note that ${\log }_{10}60$ and ${\log }_{10}40$ are both greater than 1, but ${\log }_{10}60$ is greater than ${\log }_{10}40$.
4. Step 4: Consider the denominators: ${\log }_{10}4$ and ${\log }_{10}3$. Since 4 is greater than 3, ${\log }_{10}4$ is greater than ${\log }_{10}3$.
5. Step 5: Conclude that the fraction with the larger numerator and smaller denominator will be greater. Therefore, compare the two expressions to determine which is larger.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for the exponent in equations of the form b^y = x, where b is the base. The logarithm log_b(x) answers the question: to what power must the base b be raised to obtain x? Understanding how to manipulate and compare logarithmic expressions is essential for solving problems involving logarithms.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, facilitating comparisons between them. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when comparing logarithms with different bases, as it enables us to express them in terms of a common base, typically base 10 or base e.

Inequalities and Comparison of Values

Understanding how to compare values, especially in the context of inequalities, is crucial for determining which of two logarithmic expressions is greater. This involves analyzing the growth rates of the logarithmic functions and their respective arguments. By applying properties of logarithms and inequalities, we can derive conclusions about the relative sizes of log4(60) and log3(40) without direct computation.