Sec 3 A-Math Exponential & Logarithmic Functions topical questions

Solve the equation lg x = −2.

Simplify lg 5 + lg 8 − lg 4.

Write the statement log₅ 25 = 2 in index form.

Evaluate ln e⁴ and lg 10.

Solve the equation 5^(2x + 1) = 125.

Solve the equation 3ˣ = 20, giving your answer correct to 3 significant figures.

Solve the equation log₂(2x + 1) = 3.

1. (a)Write the equation in index form.[2]
2. (b)Solve the equation.[2]
3. (c)Confirm that your answer is valid.[1]

Simplify 2 lg 5 + lg 16 − 2, giving your answer as a single logarithm or a number.

Solve the equation 4^(x + 1) = 8ˣ.

Find the exact value of log₈ 32.

Solve the equation 2·3^(2x) − 5·3ˣ − 3 = 0.

1. (a)Using the substitution y = 3ˣ, show that 2y² − 5y − 3 = 0.[2]
2. (b)Solve the quadratic in y.[2]
3. (c)Hence solve the original equation, rejecting any impossible value.[2]

Solve the equation ln(x + 2) + ln(x − 2) = ln 5.

1. (a)Combine the two logarithms into one.[2]
2. (b)Solve the resulting quadratic equation.[2]
3. (c)State the final solution, rejecting any invalid candidate.[2]

A car loses half its value every 3 years. Its value, V dollars, t years after purchase is modelled by V = 180000 × (1/2)^(t/3).

1. (a)Find the value of the car after 9 years.[2]
2. (b)Find the age of the car when its value is 45 000 dollars.[3]
3. (c)Interpret the constant 3 in the model.[1]

A quantity y is modelled by y = A·e^(kx), where A and k are constants. When x = 0, y = 6, and when x = 2, y = 18.

1. (a)Find the value of A.[2]
2. (b)Show that e^(2k) = 3.[2]
3. (c)Find the value of k, correct to 3 significant figures.[2]

It is given that lg 2 = 0.301 and lg 3 = 0.477, correct to 3 decimal places. Using only these values and the laws of logarithms:

1. (a)Find lg 12.[3]
2. (b)Find lg 5.[3]