Logarithms are a fundamental concept in mathematics, particularly in algebra and calculus. A logarithm is the power to which a number, called the base, must be raised to obtain another number. Understanding logarithms is essential for solving exponential equations, growth decay problems, and in various scientific applications.

## 1. If log10 2 = 0.3010, find the value of log10 5.

a) 0.6989

b) 0.6990

c) 0.7000

d) 0.7010

### Answer:

a) 0.6989

### Explanation:

```
Using the property log10(a/b) = log10 a - log10 b, log10 5 = log10(10/2) = log10 10 - log10 2 = 1 - 0.3010 = 0.6989.
```

## 2. Find the value of log2 8.

a) 2

b) 3

c) 4

d) 5

### Answer:

b) 3

### Explanation:

```
log2 8 = x implies 2^x = 8. Since 2^3 = 8, x = 3.
```

## 3. If log3 (9x) = 4, find x.

a) 27

b) 36

c) 45

d) 81

### Answer:

d) 81

### Explanation:

```
log3 (9x) = 4 implies 3^4 = 9x.
81 = 9x, thus x = 81 / 9 = 9.
```

## 4. Calculate log5 125.

a) 2

b) 3

c) 4

d) 5

### Answer:

b) 3

### Explanation:

```
log5 125 = x implies 5^x = 125. Since 5^3 = 125, x = 3.
```

## 5. If log x = 0.3010, find the value of x.

a) 2

b) 5

c) 10

d) 20

### Answer:

a) 2

### Explanation:

```
log x = 0.3010 implies x = 10^0.3010. Since log10 2 ≈ 0.3010, x ≈ 2.
```

## 6. Evaluate log10 1000.

a) 1

b) 2

c) 3

d) 4

### Answer:

c) 3

### Explanation:

```
log10 1000 = log10(10^3) = 3.
```

## 7. Find the value of log4 64.

a) 2

b) 3

c) 4

d) 5

### Answer:

b) 3

### Explanation:

```
log4 64 = x implies 4^x = 64. Since 4^3 = 64, x = 3.
```

## 8. If log2 x = 5, find x.

a) 16

b) 32

c) 64

d) 128

### Answer:

b) 32

### Explanation:

```
log2 x = 5 implies x = 2^5 = 32.
```

## 9. What is the value of log10(1/100)?

a) -1

b) -2

c) -3

d) -4

### Answer:

b) -2

### Explanation:

```
log10(1/100) = log10(10^-2) = -2.
```

## 10. Find log7(1/49).

a) -1

b) -2

c) -3

d) -4

### Answer:

b) -2

### Explanation:

```
log7(1/49) = log7(7^-2) = -2.
```

## 11. If log3 x – log3 27 = 2, find x.

a) 81

b) 243

c) 729

d) 2187

### Answer:

b) 243

### Explanation:

```
log3 x - log3 27 = 2 implies log3 x = log3 27 + 2.
x = 3^(3+2) = 3^5 = 243.
```

## 12. Calculate log2(1/32).

a) -3

b) -4

c) -5

d) -6

### Answer:

c) -5

### Explanation:

```
log2(1/32) = log2(2^-5) = -5.
```

## 13. What is the value of log(0.0001)?

a) -2

b) -3

c) -4

d) -5

### Answer:

c) -4

### Explanation:

```
log(0.0001) = log(10^-4) = -4.
```

## 14. Find log5 25.

a) 1

b) 1.5

c) 2

d) 2.5

### Answer:

c) 2

### Explanation:

```
log5 25 = log5(5^2) = 2.
```

## 15. If log(2x) = 3, find x.

a) 500

b) 1000

c) 1500

d) 2000

### Answer:

b) 1000

### Explanation:

```
log(2x) = 3 implies 2x = 10^3.
x = 1000 / 2 = 500.
```

## 16. Evaluate log2 16.

a) 2

b) 3

c) 4

d) 5

### Answer:

c) 4

### Explanation:

```
log2 16 = log2(2^4) = 4.
```

## 17. If log x + log y = log 50 and x*y = 50, find x and y.

a) x = 5, y = 10

b) x = 10, y = 5

c) x = 25, y = 2

d) x = 2, y = 25

### Answer:

b) x = 10, y = 5

### Explanation:

```
log x + log y = log 50 implies x*y = 50. Possible pairs (x, y) are (1, 50) and (10, 5). The pair (10, 5) satisfies both conditions.
```

## 18. Simplify: log(100/10).

a) 1

b) 2

c) 0

d) 10

### Answer:

a) 1

### Explanation:

```
log(100/10) = log(10) = 1.
```

## 19. If log4(x^2) = 4, find x.

a) 4

b) 8

c) 16

d) 32

### Answer:

b) 8

### Explanation:

```
log4(x^2) = 4 implies x^2 = 4^4.
x^2 = 256, thus x = √256 = 16.
```

## 20. Calculate log2 1.

a) 0

b) 1

c) 2

d) 3

### Answer:

a) 0

### Explanation:

```
log2 1 = 0, as 2^0 = 1.
```