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Questions and Answers
Which of the following functions does NOT exhibit exponential decay?
Which of the following functions does NOT exhibit exponential decay?
- $i(x) = 2^x$ (correct)
- $f(x) = 3x$ (correct)
- $h(x) = 2^{-x}$
- $g(x) = \frac{1}{4x}$
How can the equation $y = \log_3 x$ be expressed in exponential form?
How can the equation $y = \log_3 x$ be expressed in exponential form?
- $y^3 = x$
- $3^y = x$ (correct)
- $y = 3^x$
- $x^3 = y$
What is the value of $\log_5 625$?
What is the value of $\log_5 625$?
- 4 (correct)
- 3
- 6
- 5
If $\log_2 64 = x$, what is the value of x?
If $\log_2 64 = x$, what is the value of x?
Which of the following logarithmic equations is correctly transformed into exponential form?
Which of the following logarithmic equations is correctly transformed into exponential form?
What is the result of applying the division rule to the expression $a^{10} / a^{6}$?
What is the result of applying the division rule to the expression $a^{10} / a^{6}$?
When applying the power of a power rule to $a^{7^{2}}$, what is the correct simplification?
When applying the power of a power rule to $a^{7^{2}}$, what is the correct simplification?
What is the outcome of the expression $a^{0}$ regardless of the value of $a$?
What is the outcome of the expression $a^{0}$ regardless of the value of $a$?
How would you express the negative exponent $a^{-7}$ in a different form?
How would you express the negative exponent $a^{-7}$ in a different form?
Which rule applies to the expression $(ab)^{4}$ to distribute the exponents?
Which rule applies to the expression $(ab)^{4}$ to distribute the exponents?
What is the simplified form of $a^{3}/b^{3}$ using the power of a fraction rule?
What is the simplified form of $a^{3}/b^{3}$ using the power of a fraction rule?
For the fractional exponent $a^{3/5}$, how would it be expressed in radical form?
For the fractional exponent $a^{3/5}$, how would it be expressed in radical form?
In the exponential function $f(x) = b^{x}$, which condition must the base $b$ satisfy?
In the exponential function $f(x) = b^{x}$, which condition must the base $b$ satisfy?
What is the inverse of the function $f(x) = 2x - 5$?
What is the inverse of the function $f(x) = 2x - 5$?
Which of the following statements correctly describes an inverse function?
Which of the following statements correctly describes an inverse function?
What is the inverse of the function $f(x) = 3x + 6$?
What is the inverse of the function $f(x) = 3x + 6$?
If $f(x) = 2x - 4$, which of the following pairs is not on the graph of $f$?
If $f(x) = 2x - 4$, which of the following pairs is not on the graph of $f$?
Using the multiplication rule of exponents, what is the result of $m^7 m^3$?
Using the multiplication rule of exponents, what is the result of $m^7 m^3$?
What is the correct approach to finding the inverse of a function?
What is the correct approach to finding the inverse of a function?
How do you determine if two functions are inverses of each other?
How do you determine if two functions are inverses of each other?
Which equation represents the function for $f(x) = 3x + 6$ on the graph?
Which equation represents the function for $f(x) = 3x + 6$ on the graph?
What is an exponential equation?
What is an exponential equation?
What form does an exponential growth function take?
What form does an exponential growth function take?
Which of the following describes exponential decay?
Which of the following describes exponential decay?
In the context of exponential functions, what does the constant k represent?
In the context of exponential functions, what does the constant k represent?
Which of the following scenarios demonstrates exponential growth?
Which of the following scenarios demonstrates exponential growth?
What does the power rule for logarithms state regarding expressions of the form $\log_b(M^N)$?
What does the power rule for logarithms state regarding expressions of the form $\log_b(M^N)$?
How does the division rule for logarithms simplify the expression $\log_b(X/Y)$?
How does the division rule for logarithms simplify the expression $\log_b(X/Y)$?
Which of the following correctly expands the expression $\log_5(a^2 b^3 c^4)$?
Which of the following correctly expands the expression $\log_5(a^2 b^3 c^4)$?
Using the multiplication rule, how can $\log_4(XY)$ be expressed?
Using the multiplication rule, how can $\log_4(XY)$ be expressed?
What expression results from expanding $\log_3(5^2)$ using the power rule?
What expression results from expanding $\log_3(5^2)$ using the power rule?
What is the result when applying the division rule to the expression $\log_5(15) - \log_5(5)$?
What is the result when applying the division rule to the expression $\log_5(15) - \log_5(5)$?
When simplifying $3 \log_4(w) + 2 \log_4(x) - 3 \log_4(y) + 4 \log_4(z)$ into a single logarithm, what is the final expression?
When simplifying $3 \log_4(w) + 2 \log_4(x) - 3 \log_4(y) + 4 \log_4(z)$ into a single logarithm, what is the final expression?
What does the expansion of $\log_4(xy^2)$ result in?
What does the expansion of $\log_4(xy^2)$ result in?
If $\log_3(A) + \log_3(B)$ simplifies to which expression using the multiplication rule?
If $\log_3(A) + \log_3(B)$ simplifies to which expression using the multiplication rule?
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Study Notes
Inverse Functions
- An inverse function is a function that reverses the effect of another function.
- If f is a function, its inverse is denoted by f-1.
- For a function and its inverse, f(f-1(x)) = x and f-1(f(x)) = x.
- To find the inverse of a function, switch the x and y variables and solve for y.
Laws of Exponents
- Multiplication Rule: ax * ay = ax + y
- Division Rule: ax / ay = ax - y
- Power of a Power Rule: (ax)y = axy
- Power of a Product Rule: (ab)x = ax * bx
- Power of a Fraction Rule: (a/b)x = ax / bx
- Zero Exponent: a0 = 1
- Negative Exponent: a-x = 1 / ax
- Fractional Exponent: ax/y = √yax
Exponential Functions
- An exponential function is a function of the form f(x) = bx where b > 0 and b ≠1.
- To evaluate an exponential function, substitute the value of the variable into the function.
Exponential Equations
- Exponential equations are equations where the exponent is a variable.
- To solve an exponential equation, express both sides of the equation as powers with the same base, then solve the resulting equation.
Application of Exponential Functions
-
Exponential Growth: Describes a process where a quantity increases at an ever-increasing rate over time.
-
Exponential Decay: Describes a process where a quantity decreases at a consistent percentage rate over time.
-
Examples of Exponential Growth:
- A colony of bacteria in a human body
- The population of the Philippines
- Compound Interest
-
Examples of Exponential Decay:
- The concentration of medicine in a patient’s body over time
- The value of gadgets over time
- The remains of a radioactive element
Logarithmic Functions
- A logarithmic function is a function of the form y = logb x where b > 0, b ≠1 and x > 0.
- y = logb x is equivalent to x = by.
Laws of Logarithms
- Multiplication Rule: logb (MN) = logb M + logb N
- Division Rule: logb (M/N) = logb M - logb N
- Power Rule: logb (MN) = N logb M
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