Problem 13 - Precalculus with Calculus Previews

Question

In Problems \(13-18\), find the exact value of the given logarithm. $$ \log _{10}(0.0000001) $$

Step-by-Step Solution

Answer

The exact value of \(\log_{10}(0.0000001)\) is -7.

Step 1: Convert to Exponential Form

To find the logarithm of a number, we convert it to exponential form. Here, we want to express 0.0000001 as 10 raised to some power.

Step 2: Identify the Power of 10

Notice that 0.0000001 can be written as a decimal power of 10. Specifically, 0.0000001 is equal to \(10^{-7}\).

Step 3: Apply Logarithm Property

Use the property of logarithms that states \(\log_{b}(b^x) = x\). For our problem, this becomes \(\log_{10}(10^{-7}) = -7\).

Key Concepts

Exponential Form Conversion

Converting numbers to their exponential form can be a game-changer when solving logarithms. It all starts with thinking about what it means to raise a number to a power. In this concept, you're expressing any given number as a base raised to an exponent. This is particularly useful in logarithmic problems where you need to assess the relationship between numbers.
For example, consider 0.0000001. It's just another way to express a power of 10. By understanding this, you can convert it to exponential form as follows:

• Realize 0.0000001 is a fraction less than one.
• Count the decimal places after the zero till you reach 1, which is seven places.
• You can then represent 0.0000001 as \(10^{-7}\) because you effectively moved the decimal to the left, taking 10 to the power of -7.

This conversion makes it so much easier to apply logarithmic properties.

Logarithmic Properties

Properties of logarithms are vital in simplifying and solving logarithmic expressions. One of the most straightforward properties is the identity property: \(\log_b(b^x) = x\). This means if you have the same base for a number in exponential form and the base of the logarithm, you can directly find the exponent.
Let's see this property in action with our example problem. After expressing 0.0000001 as \(10^{-7}\), applying this property is straightforward:

• The logarithm, \(\log_{10}(0.0000001)\), becomes \(\log_{10}(10^{-7})\).
• According to the property \(\log_b(b^x) = x\), this simplifies neatly to -7.

With this, you determine that \(\log_{10}(0.0000001)\) has a value of -7, showing the power of this property in simplifying logarithmic expressions.

Power of 10 Identification

Identifying the power of 10 in decimal numbers is fundamental in solving logarithmic tasks efficiently. Recognizing decimal structures like 0.0000001 quickly as powers of ten requires understanding how many times you shift the decimal point to convert into a standard power expression.
Here's how you identify the power of 10:

• Count the number of decimal places from the left up to the first non-zero digit.
• For each place moved, you increase the negative exponent of 10.
• In our example, shifting the decimal point seven places to the right gives \(10^{-7}\).

Grasping this will simplify many logarithmic calculations by reducing complex-looking decimals to simpler exponential forms.