Problem 22 - Technical Mathematics with Calculus

Question

Find the number whose common logarithm is given. $$1.584$$

Step-by-Step Solution

Answer

The number whose common logarithm is 1.584 is approximately \(10^{1.584} \approx 38.3118684955729\).

Step 1: Understanding the Problem

The exercise is asking to find the number that corresponds to a given common logarithm. A common logarithm is a logarithm with a base of 10. So, the equation we are trying to solve is \(\log_{10}(x) = 1.584\).

Step 2: Converting the Logarithmic Equation to Exponential Form

To find the value of \(x\), we need to convert the logarithmic equation into an exponential equation using the property that if \(\log_{b}(a) = c\) then \(b^c = a\). Applying this property, we get \(10^{1.584} = x\).

Step 3: Calculating the Number

Use a calculator to find the power of 10 raised to 1.584. The result will give us the value of \(x\).

Key Concepts

Logarithmic to Exponential Form

Converting logarithmic equations to exponential form is a key skill when dealing with logarithms. In the case of a common logarithm, where the base is 10, this conversion can be easily mastered. Take the example of finding a number when its common logarithm is known to be 1.584. To convert the logarithmic expression \(\log_{10}(x) = 1.584\) to its equivalent exponential form, we apply the fundamental property of logarithms: if \(\log_{b}(a) = c\), then it can be written as \(b^c = a\).

So, for our problem, we rewrite \(\log_{10}(x) = 1.584\) as \(10^{1.584} = x\). This is now in exponential form, where 10 is the base, 1.584 is the exponent, and \(x\) is the number we're trying to find. Understanding this conversion provides a clear path to solve many logarithmic problems, making the unknown number accessible through the simplicity of the exponential equation.

Solving Logarithmic Equations

When solving logarithmic equations, it's beneficial to familiarize oneself with the idea that every logarithmic statement has an equivalent exponential statement. The equivalence between logarithmic and exponential forms serves as a bridge to find the unknown value. After converting a logarithm to its exponential form—as demonstrated in the prior section with the equation \(10^{1.584} = x\)—the next step is to actually solve for \(x\).

Typically, this involves some computational work. For instance, using a scientific calculator, you would simply calculate the value of \(10^{1.584}\). The result you'll find, rounded to appropriate significance, is the number that corresponds to the common logarithm you started with. The ease of using a calculator underscores the importance of understanding how to set up the equation properly before solving it.

Base 10 Logarithm

The common logarithm, also referred to as the base 10 logarithm, is a special type of logarithm where the base is always 10. It is denoted as \(\log(x)\) without explicitly showing the base. In everyday scenarios, such as in scientific calculations or when using decibels in sound intensity measurements, common logarithms are quite prevalent.

The power of the base 10 logarithm comes from its innate association with our decimal number system. Each increase by 1 in a common logarithmic value represents a tenfold increase in the actual number. Understanding this scaling can help to conceptually grasp large numbers and can aid in functions like reading pH levels in chemistry, which are also based on common logarithms. Recognizing the importance of the base 10 logarithm not only aids in solving equations but also connects to broader applications in science and engineering.