A quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

What is a common logarithm?

A logarithm to the base of 10, and it is written as log(x) meaning log(x) with a base of 10.

If you see a log without the base shown, it is automatically understood as what?

The base of 10, also known as a common logarithm.

What is a natural logarithm?

A logarithm to the base e (2.71828…), and it is written as ln(x).

If using your calculator and you push the log button, what does it assume?

The calculator assumes that log is a common log with the base of 10.

If using your calculator and you push the ln button, what does it mean?

The calculator's button of ln is for the use of a natural logarithm.

Since 100 = 102, then log(100) = log(102) = 2, because "log(100) = y" means "10 y = 100 = 102", so y = 2. log(100) = 2

Remember that "ln( )" means the base-e log, so "ln(e^4.5)" might be thought of as "loge(e^4.5)". The Relationship says that "ln(e^4.5) = y" means "e y = e^4.5", so y = 4.5, and: ln(e^4.5) = 4.5

Simplify log(98) by using your calculator.

Since 98 is not a nice neat power of 10 (the way that 100 is), so you must plug this into your calculator, remembering to use the "LOG" key (not the "LN" key). Answer of log(98) = 1.99122607569..., or: log(98) = 1.99, rounded to two decimal places

Simplify ln(2) by using your calculator.

Since 2 is a nice neat whole number and since e isn't, then it is unlikely that 2 is a nice neat power of e. So, you must use your calculator to get an approximate answer of ln(2) = 0.69314718056..., or: ln(2) = 0.69, rounded to two decimal places.

Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1. What is the change of base formula?

Log base b of a = Log base c of a divided by Log base c of b

5^x = 212 ln(5^x) = ln(212) xln(5) = ln(212) x = ln(212)/ln(5) ...or about 3.328, rounded to three decimal places.

What are the steps used to solve for x in the equation 7Log(3x)=15?

1) Isolate the logarithmic term before you convert the log equation to an exponential equation. Divide both sides by 7. 2) Convert the log equation to an exponential equation. 3) Divide both sides by 3.

If lime juice has a pH of 1.7, what is the concentration of hydrogen ions (in mol/L) in lime juice, to the nearest hundredth? Use the formula pH = −log[H+].

pH = −log[H+] 1.7 = −log x −1.7 = log x x = 10^-1.7 x = 0.02

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Logarithms

Common and Natural Logarithms, Change of Base, Equations and Applications

Question	Answer
What is a logarithm?	A quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
What is a common logarithm?	A logarithm to the base of 10, and it is written as log(x) meaning log(x) with a base of 10.
If you see a log without the base shown, it is automatically understood as what?	The base of 10, also known as a common logarithm.
What is a natural logarithm?	A logarithm to the base e (2.71828…), and it is written as ln(x).
If using your calculator and you push the log button, what does it assume?	The calculator assumes that log is a common log with the base of 10.
If using your calculator and you push the ln button, what does it mean?	The calculator's button of ln is for the use of a natural logarithm.
Simplify log(100).	Since 100 = 102, then log(100) = log(102) = 2, because "log(100) = y" means "10 y = 100 = 102", so y = 2. log(100) = 2
Simplify ln(e^4.5).	Remember that "ln( )" means the base-e log, so "ln(e^4.5)" might be thought of as "loge(e^4.5)". The Relationship says that "ln(e^4.5) = y" means "e y = e^4.5", so y = 4.5, and: ln(e^4.5) = 4.5
Simplify log(98) by using your calculator.	Since 98 is not a nice neat power of 10 (the way that 100 is), so you must plug this into your calculator, remembering to use the "LOG" key (not the "LN" key). Answer of log(98) = 1.99122607569..., or: log(98) = 1.99, rounded to two decimal places
Simplify ln(2) by using your calculator.	Since 2 is a nice neat whole number and since e isn't, then it is unlikely that 2 is a nice neat power of e. So, you must use your calculator to get an approximate answer of ln(2) = 0.69314718056..., or: ln(2) = 0.69, rounded to two decimal places.
Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1. What is the change of base formula?	Log base b of a = Log base c of a divided by Log base c of b
Solve 5^x = 212.	5^x = 212 ln(5^x) = ln(212) xln(5) = ln(212) x = ln(212)/ln(5) ...or about 3.328, rounded to three decimal places.
What are the steps used to solve for x in the equation 7Log(3x)=15?	1) Isolate the logarithmic term before you convert the log equation to an exponential equation. Divide both sides by 7. 2) Convert the log equation to an exponential equation. 3) Divide both sides by 3.
If lime juice has a pH of 1.7, what is the concentration of hydrogen ions (in mol/L) in lime juice, to the nearest hundredth? Use the formula pH = −log[H+].	pH = −log[H+] 1.7 = −log x −1.7 = log x x = 10^-1.7 x = 0.02

Created by: c8e109

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