1.4.5 Practice set: exponents and logarithms

Exponent rules

Exponent rules (lesson 1.4.1)

0 of 7 correct

1.
Simplify $a^5 \cdot a^{-2}$
2.
Simplify $\dfrac{(3x^2)^3}{x^4}$
3.
Evaluate $16^{3/4}$
4.
Simplify $\sqrt{x} \cdot x^{3/2}$
5.
Evaluate $(-2)^4 \cdot 2^{-3} \cdot 7^0$
6.
Evaluate $\left(\tfrac{1}{2}\right)^{-3}$
7.
Evaluate $-2^4$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Exponential functions and e

Exponentials and e (lesson 1.4.2)

0 of 6 correct

1.
Simplify $e^{3x} \cdot e^{-x}$
2.
Simplify $\left(e^{2}\right)^{x}$
3.
Write $\left(\tfrac{1}{3}\right)^x$ as a power of 3
4.
A value starts at 500 and falls 10% at each step. What is it after 2 steps?
5.
Evaluate $\left(1 + \frac{1}{n}\right)^n$ for $n = 4$ (a fraction or a decimal)
6.
A value grows 5% per step. By what factor has it grown after 10 steps? (4 decimal places, or a power)

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Logarithms

Logarithms (lesson 1.4.3)

0 of 7 correct

1.
Evaluate $\log_5 125$
2.
Evaluate $\log_2 \tfrac{1}{8}$
3.
Evaluate $\ln \sqrt{e}$
4.
Write $2\ln 3 + \ln 4$ as a single logarithm
5.
If $\ln x = a$ and $\ln y = b$ , write $\ln\left(\sqrt{x}\, y^3\right)$ in terms of $a$ and $b$
6.
Write $5^x$ as a power of $e$
7.
Evaluate $\log_4 2$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Solving equations

Solving for x (lesson 1.4.4)

0 of 7 correct

1.
Solve $2^x = 10$
2.
Solve $3e^{2x} = 48$
3.
Solve $1.05^x = 2$
4.
Solve $e^{x} = 3e^{-x}$
5.
Solve $\log_2(x - 3) = 4$
6.
Solve $e^{2x} - 4e^x - 5 = 0$
7.
Solve $\ln x + \ln(x + 3) = \ln 10$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Solve for an unknown exponent

Write solve_exponent(a, b, c) that returns the $x$ solving $a \cdot b^x = c$ .

Isolate the power first, then use logs, as in lesson 1.4.4.
Raise ValueError when there is no solution: when b <= 0, when b == 1, when a == 0, or when c / a is not positive.

The tests check easy cases, a decay base below 1, a negative a with a negative c, and each kind of impossible equation.

See the tests

import math


def _raises_value_error(a, b, c):
    try:
        solve_exponent(a, b, c)
    except ValueError:
        return True
    return False


def test_simple_power():
    """2^x = 8 gives x = 3"""
    expect_close(solve_exponent(1, 2, 8), 3.0, what="solve_exponent(1, 2, 8)")


def test_isolate_first():
    """5 · 2^x = 40 gives x = 3 (divide by 5 first)"""
    expect_close(solve_exponent(5, 2, 40), 3.0, what="solve_exponent(5, 2, 40)")


def test_needs_logs():
    """100 · 1.05^x = 200 gives x = ln 2 / ln 1.05"""
    expect_close(solve_exponent(100, 1.05, 200), math.log(2) / math.log(1.05), what="solve_exponent(100, 1.05, 200)")


def test_decay():
    """A base below 1 works too: 2 · 0.5^x = 0.25 gives x = 3"""
    expect_close(solve_exponent(2, 0.5, 0.25), 3.0, what="solve_exponent(2, 0.5, 0.25)")


def test_answer_checks_out():
    """Putting the answer back in gives c: 3 · 7^x = 50"""
    x = solve_exponent(3, 7, 50)
    expect_close(3 * 7**x, 50.0, what="3 * 7**x with x = solve_exponent(3, 7, 50)")


def test_negative_a_and_c():
    """-2 · 3^x = -18 gives x = 2: only the sign of c / a matters"""
    expect_close(solve_exponent(-2, 3, -18), 2.0, what="solve_exponent(-2, 3, -18)")


def test_negative_target_has_no_solution():
    """2^x = -8 has no solution, so raise ValueError"""
    assert _raises_value_error(1, 2, -8), "expected ValueError for 1 · 2^x = -8"


def test_base_one_has_no_solution():
    """1^x is always 1, so 1^x = 5 raises ValueError"""
    assert _raises_value_error(1, 1, 5), "expected ValueError for base 1"


def test_zero_a_has_no_solution():
    """0 · 2^x is always 0, so 0 · 2^x = 5 raises ValueError"""
    try:
        solve_exponent(0, 2, 5)
    except ValueError:
        return
    except ZeroDivisionError:
        raise AssertionError("got ZeroDivisionError for a = 0: check a == 0 first and raise ValueError")
    raise AssertionError("expected ValueError for 0 · 2^x = 5")

Exponent rules

Exponential functions and e

Logarithms

Solving equations

Coding exercise

Free further reading