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Properties of Logarithms

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log_a(a) = 1
What does the identity property of logarithms state?
Type I: a^(log_a(x)) = x
What does the inverse property of logarithms state?
log_a(b^c) = c*log_a(b)
What does the exponent-to-constant property of logarithms state?
log_a(uv) = log_a(u) + log_a(v)
What does the product property of logarithms state?
log_a(u/v) = log_a(u) − log_a(v)
What does the quotient property of logarithms state?
The log is equal to 0.
If the argument of a log of any base is one...
{1, 10}
log^2(x) = log(x)
1.) Collect all the logarithmic terms on one side.
What are the steps to solve a logarithmic equation where where some terms in the equation have logarithms and some do not?
Half Life
N(t) = n° • e^ -kt
Continuous compound interest
A(t) = p • e ^rt
Product property
logb (mn) = logb (m) + logb (n)
Quotient property
logb (m/n) = logb (m) - logb (n)
Power property
logb (a) ^ P = P•logb (a)
Inverse Property
The sum of a number and its opposite is 0.
Change of base
logb X= log (x)/ log (b)
logb 1 = 0
because b^0 = 1
logb b = 1
because b^1 = b
logb b^y = y
because b^y = b^y
b^(logb^x) = x
because logb x = logb x
logb 3^e+1 = e+1
because the exponent always equals the equation.
logb u^v = Vlogb U
because the exponent can be pulled out and brought to the front
Solve logx 3
x = 10^3 = 1000
Solve log2 x = 5
x = 2^5 = 32
ln 1 = 0
because e^0 = 1
e^1 = e
ln e = 1
ey = ey
ln e^y = y
e^lnx = x
because the exponent is always the answer. ln x = ln x
Solve ln e^5
e^5 = 5
Solve e^ln4
The answer is always the exponent. ln4 = ln4
if b^u = b^v, u=v
What is the 1 to 1 property?
x = 5/3
Solve 9^2x = 3^x+5
Solve for 5^x = 30
Can't use 1 to 1 because you cannot make bases the same.
3 + 2e^2x = 50
Solve for 3 + 2e^2x = 50
They are inverses!
What is the relationship between Logs and Exponential?
Convert to Exponential.
Parent of y = log3^x
Make points of Y.
When graphing Logs, what do you start with after finding the parent Exponential?
What do you do with y points?
Plug them into the parent Exponential and find X
Only the Range!
Which of the characteristics of a graph (Domain, Rang, Increasing, Decreasing, Asymptote) rely on Y coordinates?
Arithmetic Mean
Geometric Mean
Perfect Cubes
Exponential Equation
Half Life
addition property
If a>b, then a+c>b+c
subtraction property
If a>b, then a-c>b-c
Exponential Property
log(xⁿ) = n logx
division property
If a>b and c>0, then a/c>b/c
log28= x
Now, rewrite the logarithmic equation as an exponential equation.
change of base formula
logby=log base b of y = log y/ log b
natural log
A log of base e is called the
When the base of a logarithm is the same as its argument, the solution will always be 1.
Since the base and the argument are the same, the numerator and denominator will also be the same. When the numerator and denominator of a fraction are the same, the fraction simplifies to 1.
logb x = y
logb x = y ↔ by = x
equality property of logarithms
logb x = logb y ↔ x = y
This is called the Product Property of Logarithms.
If two or more logarithmic expressions with the same base are added, multiply the arguments to find the sum.
This is called the Quotient Property of Logarithms.
If two or more logarithmic expressions with the same base are subtracted, divide the first argument by the second argument to find the difference
Addition of logarithms with like bases involves the multiplication of arguments.
Subtraction of logarithms with like bases involves the division of arguments.
This is called the Power Property of Logarithms.
The exponent on the argument becomes a coefficient of the logarithmic expression. A coefficient of a logarithmic term can be moved to the exponent of its argument and vice versa.
Equality property
given an equation where two logarithmic expressions, with identical bases, are equal to each other, the arguments of those expressions are also equal.
The P represents the principle which is the same as the amount at the very beginning.
If the situation asks you to solve for the time needed to reach a given amount, consider using logarithms. It is possible to replace the A using function notation.
If 1 + r = 1, then the r = 0 and it is a linear equation!
The t represents the amount of times the growth has occurred. If a fly population doubles every year and you want the population after 10 years, then t = 10. It is important to make sure your units match.