Question

Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places. $$\log _{6} 60$$

Short answer

Answer: The value of $\log_{6}60$ in terms of the natural logarithm is approximately 2.2843.

Step-by-step solution

Apply the change of base formula

Using the change of base formula on \(\log_{6}60\), we get: $$\log_{6}60 = \frac{\log_{e}60}{\log_{e}6}$$

Use calculator to find the logarithm values

To find the value of the logarithm, we will now calculate the natural logarithm of \(60\) and \(6\) using a calculator (most calculators have a built-in ln button) and then divide them: $$\log_{e}60 ≈ 4.0943$$ $$\log_{e}6 ≈ 1.7918$$

Divide the two logarithm values

Now, we will divide the two calculated logarithm values in order to find the value of the initial logarithm: $$\frac{\log_{e}60}{\log_{e}6} ≈ \frac{4.0943}{1.7918} ≈ 2.2843$$

Round the result to four decimal places

Lastly, round the result to four decimal places: $$\log_{6}60 ≈ 2.2843$$ So, the value of the logarithm in terms of the natural logarithm is approximately \(2.2843\).

Key concepts

Natural Logarithm

The natural logarithm is a special logarithm where the base is the mathematical constant e, approximately equal to 2.71828. In mathematical notation, the natural logarithm of a number x is denoted as \(\ln(x)\) or sometimes \(\log_e(x)\), where e represents Euler's number. This type of logarithm is significant in various fields of science and mathematics because it describes growth processes and comes up naturally in the context of calculus due to its unique properties.

For instance, the natural logarithm is the inverse function of exponentiation with base e. This means that if \(e^y = x\), then \(\ln(x) = y\). It plays a central role in solving equations that involve exponential growth or decay and is pivotal in calculus for finding derivatives and integrals involving exponential functions. When logarithms with different bases need to be calculated, the natural logarithm can be used as a mediator through the change of base formula, simplifying computations.

Logarithmic Calculations

Logarithmic calculations can seem daunting at first, but once you understand the properties and laws governing logarithms, they become much more manageable. Some of these key properties include the product, quotient, and power rules, which allow for the simplification of logarithmic expressions.

Using a calculator, you can find the logarithm of a number with any base by employing the change of base formula. This formula states that \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\), where \(\log_b(a)\) is the logarithm of a with base b, and \(\ln\) denotes the natural logarithm. This formula allows you to perform logarithmic calculations for any base using the natural logarithm function, which is commonly available on scientific calculators. For practical reasons, when the base is not 10 or e, the change of base formula is extremely helpful, as it opens up the possibility of calculating otherwise complex logarithms with ease.

Rounding Decimal Places

Rounding decimal places is an important skill in mathematics, especially when dealing with irrational numbers or the results of complex calculations. When rounding to a specific number of decimal places, you look at the digit right after your rounding threshold. If this digit is 5 or above, you round the last retained digit up by one. If it is below 5, you leave the last retained digit as it is.

For instance, when we calculated \(\frac{\ln(60)}{\ln(6)}\) and got an approximate value of 2.28427, we would round it to 2.2843 if we're rounding to four decimal places. This is done to make the number easier to read and handle, especially in applied settings where extreme precision is not required. Nevertheless, it's always important to keep in mind the context in which the rounding is taking place since the more you round, the more precision you potentially lose, which can be crucial in some scientific computations.