Logarithmic Functions Topic 1: Evaluating Logarithmic Equations

I can express a logarithmic equation as an exponential equation and vice versa. I can determine the value of a logarithmic expression, such as, without technology. I can determine the value of a logarithmic expression, such as, with technology. I can solve problems that involve logarithmic scales, such as the Richter scale and the pH scale.

Explore… Logarithms are an alternate way of expressing exponents. The table below shows the relationship between exponential and logarithmic forms. Study the completed first three rows. Complete the table by filling in the blanks. Try this on your own first!

You Should Notice… There is a pattern to follow when converting between exponential and logarithmic form.

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Example 1 Convert the exponential equations into logarithmic equations, and the logarithmic equations into exponential equations. a) b) c) d) Converting between exponential and logarithmic forms

Example 2 Evaluate each logarithmic expression without using technology. a) b) c)d) Evaluating and interpreting a logarithmic expression without technology = x Since 2 to the power of 4 gives us 16, x = 4. = x Since the answer is unknown, we can use x as a placeholder to represent the answer. Then we can change the expressions to exponential form. Last, we use reasoning to figure out x’s value. Since 4 to the power of 3 gives us 16, x = 3.

Example 2 e) f) = x Since the answer is unknown, we can use x as a placeholder to represent the answer. Then we can change the expressions to exponential form. Last, we use reasoning to figure out x’s value. There is no number that can, when placed as an exponent on 2, give an answer of -4. No Solution.

Example 3 Evaluate each logarithmic equation without using technology. a) b) c) d) Solving a logarithmic equation without technology Start by changing the expressions to exponential form. If it is easy to calculate, do that. x = 7 In cases like these last two, you can take the root of both sides to isolate x.

Example 4 Describe how to estimate the value following without using technology. a) b) Estimating the value of a logarithmic expressions without technology Since the answer is unknown, we can use x as a placeholder to represent the answer. Then we can change the expressions to exponential form. Last, we use reasoning to figure out an approximate value for x. = x A good approximate answer is x = 3.9.A good approximate answer is x = 2.6.

Example 5 Evaluate each logarithmic expression using technology. a) b) Evaluating and interpreting a logarithmic expression with technology To calculate these in your calculator, use the change of base formula. Keep in mind that on most calculators, the log button is base 10.

Example 6 Using logarithms to solve a contextual problem

Example 7 The pH, is defined by the equation pH =  log C, where the concentration of hydrogen ions, C, in a solution is measured in moles per litre (mol/L). Using logarithms to determine and compare pH levels

Example 7 a)The hydrogen ion concentration, C, of a solution is mol/L. Calculate the pH of the solution. b) Use the pH scale shown in the diagram above to calculate the hydrogen ion concentration of lemon juice. Looking at the table, Divide both sides by -1. Change to exp. form.

Example 7 c)A common ingredient in cola drinks is phosphoric acid, the same ingredient found in many rust removers. In terms of hydrogen ion concentration, how much more acidic is cola, with a pH of 2.5, than milk, with a pH of 6.6? Round your answer to the nearest tenth. Since we are comparing the cola to the milk, set up a ratio that has the cola concentration on the top and the milk concentration on the bottom. MilkCola Compare cola to milk: Cola is times more acidic than milk.

Example 7 d) In terms of hydrogen ion concentration, how many times more acidic is seawater than a soapy water solution? Since we are comparing the sea water to the soapy water, set up a ratio that has the sea water concentration on the top and the soapy water concentration on the bottom. Soapy waterSea water Compare cola to milk: Sea water is times more acidic than soapy water.

Need to Know

To verify your value of a logarithm, substitute your answer into the original equation,. You can estimate the value of a logarithm of the form by converting x to a power with base b that has approximately the same value as x, then solving for y.

Need to Know Many real-life situations have values that vary greatly. A logarithmic scale with powers of 10 can be used to make comparisons between large and small values more manageable. Three examples of logarithmic scales are: Richter Scale – used to measure the magnitude of an earthquake pH Scale – used to measure acidity of a solution Decibel Scale – used to measure sound level You’re ready! Try the homework from this section.