Question

Find the number whose common logarithm is given. $$0.366$$

Short answer

\(10^{0.366} \approx 2.319\). The number whose common logarithm is 0.366 is approximately 2.319.

Step-by-step solution

Understanding the Common Logarithm

The common logarithm (base 10) of a number x is the power to which the base 10 must be raised to obtain the number x. Mathematically, if \(\log_{10}x = y\), then \(10^y = x\). In this case, we have \(\log_{10}x = 0.366\). We need to rewrite this in exponential form to solve for x.

Converting Logarithmic Form to Exponential Form

The logarithmic equation \(\log_{10}x = 0.366\) can be converted to its exponential form by raising the base 10 to the power of the given logarithm, which gives us \(10^{0.366} = x\).

Calculating the Number

We can now calculate the value of \(x\) by raising 10 to the power of 0.366, which can be done using a scientific calculator or a computational software.

Key concepts

Logarithmic to Exponential Form

Understanding how to convert from logarithmic form to exponential form facilitates the process of calculating and solving logarithmic equations. Let's delve into an example to clarify this conversion. Consider the expression \(\log_{10}x = 0.366\) where the base of the logarithm is 10, also known as the common logarithm. To find the number \(x\), you'll first need to convert this logarithmic equation into its equivalent exponential form.

The rule of conversion is straightforward. Here's how to translate it: the base 10 raised to the power of the right-hand side of the logarithmic equation equals \(x\). Therefore, \(10^{0.366} = x\). To remember this, it might help to think of logarithms as the 'opposite' of exponentiation. In the exponential equation, 10 is the base, 0.366 is the exponent, and \(x\) is the result of that exponential expression.

Calculating Powers of 10

Raising 10 to any power is a common calculation in mathematics, especially when dealing with common logarithms. The expression \(10^n\), where \(n\) is any real number, represents 10 raised to the power of \(n\). In our specific case, we're interested in \(10^{0.366}\).

To calculate this, you can use a scientific calculator by typing 10, pressing the exponent button, and finally entering 0.366. The result is equivalent to the number whose common logarithm was given in the problem. The value of powers of 10 can also be understood in terms of magnitude. For instance, \(10^1=10\), \(10^2=100\), and so on, showing increases in magnitude. As the exponent decreases towards zero, the value approaches 1 (since \(10^0=1\)) and for negative exponents, the value represents fractions (as \(10^{-1}=0.1\)).

Solving Logarithmic Equations

To solve logarithmic equations, you need to isolate the variable of interest. In our case with the equation \(\log_{10}x = 0.366\), we want to solve for \(x\). After converting the logarithmic equation to its exponential form, you're left with \(x = 10^{0.366}\).

This is now an exponential equation that can be solved numerically. In general, solving logarithmic equations often involves these steps: convert to exponential form, perform any necessary algebra to isolate the variable, and then calculate the result using a calculator, if needed. Remember, the concept hinges on understanding how exponents and logarithms are inverses of each other, and this inverse relationship is the key to moving back and forth between the two forms to find your solution.