Question

Use your calculator to find the inverse log of the following numbers. (a) \(12.7\) (b) \(-9.4\) (c) 1

Short answer

The inverse logs for 12.7, -9.4 and 1 are \(10^{12.7}\), \(10^{-9.4}\) and \(10^{1}\) (or 10) respectively.

Step-by-step solution

- Calculate the inverse log of 12.7

To calculate the inverse log of 12.7, it's necessary to use the '10 to the power of x' function. It would therefore be like calculating \(10^{12.7}\). The result will be a very large number.

- Calculate the inverse log of -9.4

Similarly, in order to calculate the inverse log of -9.4, use the '10 to the power of x' function. Essentially, you're calculating \(10^{-9.4}\). As -9.4 is a negative number, the result will be a decimal fraction less than 1.

- Calculate the inverse log of 1

Lastly, you're calculating the inverse log of 1. Using the '10 to the power of x' function, this translates to calculating \(10^{1}\). The result would be 10.

Key concepts

Logarithm Calculations

Understanding logarithms is crucial for solving various mathematical and scientific problems. A logarithm answers the question: 'To what power do we raise a certain number (known as the base) to obtain another number?'. For example, in the expression \( \log_{10}(100) \), the base is 10 and we are asking what power we need to raise 10 to get 100, which is 2.

Inverse logarithm, conversely, is the process of finding the number that a base (commonly 10 in scientific calculations) must be raised to in order to achieve the value given as the input of the inverse log. The inverse log is often notated as \( 10^{x} \) or \( e^{x} \) for natural logarithms (where the base is \( e \) – Euler’s number). In essence, when you're asked to find the inverse log, you're being tasked with exponentiating the base by the logarithm's value.

Exponentiation

Exponentiation is a mathematical operation that involves taking one number (the base) and raising it to the power of another number (the exponent). In the general form, it's written as \( b^{n} \), where \( b \) is the base and \( n \) is the exponent. When the exponent is a positive integer, exponentiation corresponds to repeated multiplication of the base: \( b^{n} = b \times b \times ... \times b \) (n times).

However, when dealing with negative exponents, the operation implies finding the reciprocal of the base raised to the absolute value of the exponent, as in \( b^{-n} = 1/(b^{n}) \). This is why \( 10^{-9.4} \) as an inverse log produces a small decimal fraction, while a positive exponent like in \( 10^{12.7} \) results in a very large number.

Scientific Calculator Usage

A scientific calculator is an indispensable tool for students tackling complex calculations that involve operations such as exponentiation and logarithms. To find the inverse logarithm using a scientific calculator, one would typically use the button labeled with \( 10^{x} \), or sometimes a \( \text{log}^{-1} \) or \( \text{antilog} \) button.

By inputting the logarithm value and pressing this button, the calculator performs the calculation for the inverse logarithm. For instance, to solve the provided exercise, you would key in the number 12.7 and press the \( 10^{x} \) button to calculate \( 10^{12.7} \). It's important to become familiar with your own calculator's functions and sometimes refer to the manual, as not all scientific calculators are laid out the same way.

Mathematical Concepts in Chemistry

In chemistry, mathematical concepts like logarithms are pivotal in areas such as pH calculations and chemical kinetics. The pH scale is logarithmic, meaning each number on the scale represents a tenfold change in the concentration of hydrogen ions in a solution. For example, a pH of 3 is ten times more acidic than a pH of 4.

Moreover, reaction rates can involve exponential relationships, with rate constants often determined through calculations involving the natural logarithm (using Euler’s number as a base) or common logarithm (base 10). Understanding how to manipulate and interpret logarithms is essential for calculating concentrations, reaction rates, and equilibrium constants in chemistry.