Algebra 2B - Unit 2 Exam Flashcards

Solve the equation for x: $2^{4x-3} = 2^{x+18}$.

Solve the equation for x: $3^{6x+7} = 243$.

Solve the equation for x: $5 imes 10^{x+2} = 5000$.

Solve the equation for x: $4 imes 3^{x-2} = 60$.

Match each logarithm with its equivalent expression.

What is the value of $log_4 62$?

If $\frac{3^{x}}{5} = 24$, what is the value of x?

Solve the equation for x: $5^{4x-1} = 845$.

Solve $7^{2x-9} = 441$ for x.

Cara tracked the population of fish in a pond. Which equation can be used to determine the number of fish, $f$, after $t$ years if the population triples each year?

Model the population of Swanford, starting from 12,500 and decreasing at 4% each year with the equation where $p$ is the population after $t$ years.

Based on the inequality $1,000(2)^{2t} > 50,000$, when will the bacteria population be greater than 50,000?

How old is a fossil found with 75% of the original carbon-14, given the half-life is 5,730 years?

Match each part with its value for the logarithmic function f(x).

Match each part with its value for the logarithmic function f(x) = log x.

Which interval represents all of the x-values where $f(x) = log_{13} x$ is positive?

Which graph represents the function $h(x) = log_5 x$?

What is the effect on the function $f(x) = log_{3/4} x - 2$ when it transforms to $g(x) = log_{3/4} x + 4$?

Describe the transformation on $f(x)$ when it changes to $g(x) = f(-2x)$.

Which is the transformation rule and the function rule for $g(x)$ derived from $f(x) = log_3 x - 2$?

Solving Equations

For the equation (2^{4x-3} = 2^{x+18}), the solution is (x = 7).
The equation (3^{6x+7} = 243) yields (x = -\frac{1}{3}).
Solving (5 \cdot 10^{x+2} = 5000) results in (x = 1).
In the equation (4 \cdot 3^{x-2} = 60), the solution is (x = \log_3 15 + 2).
The equation (5^{4x-1} = 845) gives (x = 1.297).
Solving (7^{2x-9} = 441) results in (x = 6.065).
For ( \frac{3^x}{5} = 24), the estimated value of (x) is approximately (14.464).

Logarithmic Equivalents

( \ln 4^{15} ) can be expressed as (15 \ln 4).
( \log_m 15^4 ) is equivalent to (4 \log_m 15).
The expression ( \log 4^m ) translates to (m \log 4).

Logarithmic Values

Value of ( \log_4 62 ) is approximately (2.977).

Population Growth and Decay Models

Cara's fish population can be modeled by (f = 8 \cdot 3^{(t-1)}), indicating tripling each year.
The population decline in Swanford can be modeled with (p = 12,500(0.96)^t), representing a 4% decrease annually.
A bacterial sample starting at 1,000 doubles every half hour; it will exceed 50,000 when (t > 3) hours.

Carbon Dating

A fossil with 75% of its original carbon-14 is approximately (2,378) years old, based on a half-life of (5,730) years.

Logarithmic Functions and Graphs

For (f(x)), the x-intercept is at ((1, 0)) and a vertical asymptote occurs at (x = 0).
The intersection of (f(x)) with (y = 1) happens at ((0.75, 1)).
In the function (f(x) = \log x), the x-intercept remains at ((1, 0)), and vertical asymptote persists at (x = 0), intersecting with (y = 1) at ((10, 1)).

Function Transformations

The interval for (f(x) = \log_{13} x) to be positive is (0 < x < 1).
Transformation (g(x) = \log_5 x) produces a specific graph.
The translation of (f(x) = \log_{3/4} x - 2) to (g(x) = \log_{3/4} x + 4) results in a 6-unit upward shift.
The transformation from (f(x)) to (g(x) = f(-2x)) reflects (f(x)) across the y-axis and compresses it horizontally by a factor of (1/2).
The transformation rule for (g(x)) transitioning from (f(x) = \log_3 x - 2) is (g(x) = f(x + 2) + 3), confirming the rule as (g(x) = \log_3 (x + 2) + 1).