Characteristics of Exponential Functions
The domain of f(x)=a^x.
What is "all real numbers"? OR
What is
(-\infty, \infty)?
The range of logarithm functions.
What is all real numbers? OR
What is (-infty, infty)?
The logarithm of 1 is this, regardless of its base.
What is zero?
Simplify a^0 when a>0.
What is 1?
Solution to 4^x=64.
What is 3?
The value of "n" in the discrete compound interest model represents this.
What is "the number of compounding periods per year"?
The y-intercept of f(x)=a^x .
What is (0,1)?
The x-intercept of f(x)=log_a(x).
What is (1,0)?
If M,N>0 then log(MN) expands to this.
What is log(M) + log(N)?
What is "take the reciprocal"? OR
What is "to flip it over"?
Solution to log_2(128)=x.
What is x=7?
The value "C" in the model f(x) = A(B^x) + C
What is the horizontal asymptote?
This behavior of an exponential function whose base is between zero and one, as the input values increase.
What is decreasing?
If the base of the logarithmic function is between zero and one, then the graph is doing this as the input values increase.
What is decrease?
The -log(N) results from this rule, when M=1.
What is log(M/N)=log(M) - log(N)? OR
What is the "Quotient-to-Difference Rule"?
The transformation made by changing f(x) = a^x to g(x) = (1/a)^x
What is "negating the input"? OR
What is "reflection over the y-axis"?
Solution to e^(2x+1) = 5. Exactly, mind you.
What is (ln(5)-1)/2
In the model A(t) = 100(0.5)^(t) this would be the rate of cooling if we change the base to the natural number. Be exact.
What is ln(1/2)? OR
What is -ln(2)?
Reflect the graph of f(x)=a^x over the x-axis.
What is g(x) = -a^x?
If log_a(x)=y then the output y means this.
What is "the exponent needed on a to get x"?
Expand log((xsqrt(y))/z^3) assuming x,y,z >0 .
What is log(x)+1/2log(y)-3log(z)?
The correct terminology to change an exponential equation into an algebraic equation.
What is "to compose with its inverse"? OR
What is "to take the logarithm of both sides"?
Exact solution to ln(x-1)+6 = 10 .
What is e^4+1?
In the Newton's Law of Cooling model T(t) = 50+75e^(-0.5t) this is the initial temperature.
What is 125 units?
All bases may be changed to the natural number base through this rewriting.
What is b^x=e^(xln(b))?
Logarithms may yield the same value even if their bases are different through this formula. State the formula in its full generality.
What is log_a(M) = log_b(M)/log_b(a)?
Condense 4ln(x)-2/3ln(z)+1 into one logarithmic statement (even the constant).
What is ln((x^4e)/z^(2/3))?
The condition on the input value in order for us to conclude log_a(M^r) = rlog_a(M) and stay in the real number system.
What is M>0?
Exact solution to ln(x)+ln(x-2) = 0.
What is 1+sqrt(2)?
\lim_{n\rightarrow \infty} (1+x/n)^n
What is e^x?