Unit 2 - 3 Logarithms
What is the relationship between the graph y=a^x and y = loga^x
The graph of the exponential function y=ax is a reflection of the logarithmic graph y=loga x in the line y=x. This tells us that y=ax and y=loga x are inverse functions.
What does 2^(2x+1) equal?
(4^x)*2
What kind of a number is the logarithm of a negative number?
A complex number
What is a logarithm to the base e called, and what is it abbreviated too?
A logarithm to the base e is called a natural logarithm and is abbreviated to ln
What point do all logarithmic graphs go through, and why?
All logarithmic graphs go through the point (1,0) because when x=1 we have y=log a (1) which is equivalent to a^y=1. Since any number raised to the power of 0 is 1, y=0 when x=1 for any value of a>0.
How would you find the solution to this equation 4^x −5× 2^x +4=0?
It is a disguised quadratic, so substitue y=x^2 into the equation. Factorise to solve for y, then put 2^x = y solutions, use logs to work out x.
What does log10 (x) equal?
Log (x)
If a^x = b, then how would you rewrite it using logs?
Log a (b) = x For a > 0
What is the multiplication law of logarithms?
Log a (x) + Log a (y) = Log a (xy)
When we see just log written, what does it mean?
Log10
How can you solve this equation: 10^x = 30?
Log10(30) = x =1.48
Does log a (x+y) = log a (x) + log a (y)?
No!
How would you solve the equation 43^x−1=3^x+2, giving your solution in the form log m/log n where m and n are rational numbers?
Taking logs on both sides since the solution is to be given in terms of logs to the base 10. Using the rule log a^x=x log a on both sides Multiplying out the bracket Collecting together terms containing x on the LHS Factorising the LHS so that xonly appears one in the equation Using the laws of logarithms to combine to a single log on each side of the equation
Why is there no logarithmic graphs for negative x values?
There is no logarithmic graph for negative values of x because the base a is always positive and when we raise a positive number to a power, the result is always positive.
What is the division rule of logarithms?
log a (x) - log a (y) = log a (x/y)
What is the power rule of logarithms?
log a (x^k) = k log a (x)
For 0<a<1, if x>1, y is...
negative
if 0<x<1 then y is...
negative
For base a>1, if x>1, then y is...
positive
if 0<x<1 then y is...
positive
What is equal to log a (1/x)?
-log a (x)
What does log a (1) equal?
0
What does log a (a) equal?
1
What must the base of a log always be?
A positive integer greater than 1
What is e?
Euler's number which is an irrational number like pi.
What does loge (x) equal?
ln (x)
How would you make x the subject of the equation logx^(y−1)=3
use logx^a = b which is x^b = a so x^3 = y-1. Or put x up to the power of the equation on both sides to cancel the logs because x^logx^a = a
What does a^ loga^x equal?
x
What does e^lnx equal?
x
What does ln(e^x) equal?
x
What does loga (a^x) equal?
x