Using and Deriving

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Exercises for This Topic Index of OnLine Topics Everything for Calculus Everything for Finite Math Everything for Finite Math & Calculus Utility: Function Evaluator & Grapher Español 
Logarithms
We start by reviewing the basic definitions as in Section 2.3 of Calculus Applied to the Real World. If you like, you can also take a look at the topic summary material on logarithms.
Base b Logarithm
The base $b$ logarithm of $x,$ $\log_b x,$ is the power to which you need to raise $b$ in order to get $x.$ Symbolically,
1. $\log_b x$ is only defined if $b$ and $x$ are both positive, and $b \neq 1.$ 2. $\log_10 x$ is called the common logarithm of $x,$ and is sometimes written as $\log x.$ 3. $\log_e x$ is called the natural logarithm of $x$ and is sometimes written as $\ln x.$ Examples
Here are some for you to try 
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Example 1 Calculating Logarithms by Hand
Algebraic Properties of Logarithms
The following identities hold for any positive $a \neq 1$ and any positive numbers $x$ and $y.$

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Example 2 Using the Properties of Logarithms
Q Where do the identities come from?
A Roughly speaking, they are restatements in logarithmic form of the laws of exponents.
Here is an intuitive way of thinking about it: Since logs are exponents, this identity expresses the familiar law that the exponent of a product is the sum of the exponents.
The second logarithmic identity is shown in almost the identical way, and we leave it for you for practice.
Q Why is $\log_a(x^r) = r \log_ax$ ?Identity (d) we will leave for you to do as practice.
Q Why is $\log_a(1/x) = \log_ax$ ?Q Why is
Since logarithms are exponents, we can use them to solve equations where the unknown is in the exponent.
_{} Example 3 Solving for the Exponent
Solve the following equations for $x.$Solution We can solve both of these equations by translating from exponential form to logarithmic form.
(a) Write the given equation in logarithmic form:
$4^{x^2} = 1/64$  Exponential Form  
$\log_4(1/64) = x^2$  Logarithmic Form  
Thus,  $x^2 = \log_4(1/64) = 3$  
giving  $x = ±3^{1/2}.$ 
(b) Before converting to logarithmic form, first divide both sides of the equation by 5:
$5 (1.1^{2x+3}) = 200$  
$1.1^{2x+3} = 40$  Exponential Form  
$\log_{1.1}40 = 2x+3$  Logarithmic Form  
This gives  $2x + 3 = \ln 40/\ln 1.1 \approx 38.7039,$  Identity (e) 
so that  $x \approx 17.8520.$ 
You can now either go on and try the exericses in the exercise set for this topic.
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Exercises for This Topic Index of OnLine Topics Everything for Calculus Everything for Finite Math Everything for Finite Math & Calculus Utility: Function Evaluator & Grapher 