1.6 Exponentials and Logarithms

• If the base $a$ of an exponential function is 1, the function is a constant function.


  True
• Order the laws of exponents according to their operations:

  1️⃣ Multiplication

  2️⃣ Division

  3️⃣ Power of a Power

  4️⃣ Power of a Product

  5️⃣ Power of a Quotient
• The point (0, 1) is included on the graph of any exponential function.


  True
• The logarithmic form $y =$$\log_{a}(x)$ is equivalent to x = a^{y}</latex>, where $a$ is the base
• The formula for the power of a power law is $(a^{m})^{n} =$$a^{m \times n}$, which simplifies $(5^{2})^{3}$ to $5^{6}$, which equals 15625
• The logarithm of a quotient property is expressed as \log_{a}(\frac{x}{y}) = \log_{a}(x) - \log_{a}(y)</latex>, which shows that division inside a logarithm becomes subtraction outside
• After applying the logarithm properties, the solution for $x$ in $a^{x} =$$b$ is $x =$$\frac{\log_{a}(b)}{\log_{a}(a)}$, which simplifies to $x =$$\log_{a}(b)$ since $\log_{a}(a) =$$1$.1
• The solution for x in $a^{x} =$$b$ is $x =$$\frac{\log_{a}(b)}{\log_{a}(a)}$.


  True
• The simplified form of x\log_{2}(2)</latex> is $x$.


  True
• The base of an exponential function must be a positive real number and a \neq1
• Steps to define a logarithmic function as the inverse of an exponential function:

  1️⃣ Start with the exponential function $a^{y} =$$x$

  2️⃣ Apply the logarithm with base $a$ to both sides

  3️⃣ Define the logarithmic function as $y =$$\log_{a}(x)$
• Match the base of the exponential function with its corresponding graph behavior:

  $a > 1$ ↔️ Curves upward

  $0 < a < 1$ ↔️ Curves downward
• The equation $\log_{a}(x) =$$y$ is equivalent to $x =$$a^{y}$


  True
• The law of exponents for multiplication states that a^{m} \times a^{n} = a^{m + n}</latex>, and an example is $2^{3} \times 2^{4} =$$2^{7}$, which equals 128.
• What is $2^{3} \times 2^{4}$ simplified using the multiplication law of exponents?


  $2^{7} =$$128$
• What is $\log_{3}(\frac{27}{9})$ simplified using the quotient property of logarithms?


  $\log_{3}(3) =$$1$
• To solve a logarithmic equation, first isolate the logarithm in the equation.
• The base of an exponential function must be a positive real number and cannot equal 1.

  True
• Match the exponential form with the logarithmic form:

  x = a^{y}</latex> ↔️ $y =$$\log_{a}(x)$

  $8 =$$2^{3}$ ↔️ $\log_{2}(8) =$$3$
• What is a logarithmic function defined as the inverse of?

  Exponential function
• What is the formula for the multiplication law of exponents?

  $a^{m} \times a^{n} =$$a^{m + n}$
• What is the formula for the logarithm of a product?

  $\log_{a}(xy) =$$\log_{a}(x) +$$\log_{a}(y)$
• What is the first step in solving an exponential equation of the form $a^{x} =$$b$?


  Take logarithm of both sides
• What is the first step to solve an exponential equation $a^{x} =$$b$ using logarithms?


  Take the logarithm of both sides
• To solve $2^{x} =$$32$, take the logarithm of both sides with base 2
• Match the property with the condition for the base $a$ in an exponential function $f(x) =$$a^{x}$:


  $a > 1$ ↔️ Function increases as $x$ increases

  $0 < a < 1$ ↔️ Function decreases as $x$ increases

  $a =$$1$ ↔️ Constant function
• What are the two main components of an exponential function $f(x) =$$a^{x}$?


  Base and exponent
• If $a > 1$, the exponential function increases as x increases


  True
• A logarithmic function is the inverse of an exponential function.
• If $8 =$$2^{3}$, what is the logarithmic form?


  $\log_{2}(8) =$$3$
• Match the law of exponents with its formula:

  Power of a Power ↔️ $(a^{m})^{n} =$$a^{m \times n}$

  Power of a Product ↔️ $(ab)^{n} =$$a^{n}b^{n}$

  Power of a Quotient ↔️ $(\frac{a}{b})^{n} =$$\frac{a^{n}}{b^{n}}$
• The logarithm of a power states that $\log_{a}(x^{n}) =$$n \log_{a}(x)$


  True
• Solve $3^{x} =$$27$ for $x$.


  $x =$$3$
• Solve \log_{2}(x + 3) = 4</latex> for $x$.


  $x =$$13$
• A logarithmic function is the inverse of an exponential function.


  True
• The graph of an exponential function with $a > 1$ curves upward as $x$ increases.


  True
• The logarithmic form $y =$$\log_{a}(x)$ is equivalent to the exponential form $x =$$a^{y}$.


  True
• If $8 =$$2^{3}$, then $\log_{2}(8) =$$3$ is true.


  True
• 2^{3} \times 2^{4} = 2^{7} = 128</latex> demonstrates the multiplication law of exponents.


  True
• $\log_{2}(4 \times 8) =$$\log_{2}(4) +$$\log_{2}(8) =$$2 +$$3 =$$5$ is an example of the logarithm of a product property.


  True