Answer and explanation please

Answer:

Step-by-step explanation:

Given:

• p = log 2
• q = log 3

To find :

• log 162 in terms of p and q.

Solution:

In order to find the logarithm of 162 in terms of p and q, we can use the properties of logarithms.

We can start by expressing 162 as a product of prime factors:

Now, we can use the properties of logarithms to simplify this expression:

Since log(ab) = log(a) + log(b), we can split this into separate logarithms:

Now, we can use the fact that q = log 3:

Using the property, we get:

Now, substitute the values of p and q:

So, the logarithm of 162 in termsof p and q is:

Answer:

log 162 = 6p + 2q

Step-by-step explanation:

To write log 162 in terms of p and q, we can use the following steps:

- First, we can write 162 as a product of powers of 2 and 3, such as 162 = 2 x 3^4.

- Next, we can use the property of logarithms that log ab = log a + log b to write log 162 = log 2 + log 3^4.

- Then, we can use another property of logarithms that log a^n = n log a to write log 3^4 = 4 log 3.

- Finally, we can substitute p = log 2 and q = log 3 to get log 162 = p + 4q.

We can write 162 as follows:

```

162 = 2^6 * 3^2

```

Therefore,

```

log 162 = log (2^6 * 3^2)

```

Using the logarithmic properties of addition and multiplication, we can simplify this to:

```

log 162 = 6 * log 2 + 2 * log 3

```

Finally, substituting p = log 2 and q = log 3, we get the following expression:

```

log 162 = 6p + 2q

```

Therefore, log 162 can be written as **6p + 2q** in terms of p and q.

Okay, let's break this down step-by-step:

* log 162 = log (2^4 * 3^2)   (by prime factorization)

* log (2^4 * 3^2) = 4log2 + 2log3  (by properties of logarithms)  

* Let p = log 2 and q = log 3

* Substituting:

* log 162 = 4p + 2q

Therefore, log 162 can be written as 4p + 2q, where p = log 2 and q = log 3.

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To express log 162 in terms of p (log 2) and q (log 3), you can use logarithm properties, particularly the change of base formula. The change of base formula states that:

log_b(a) = log_c(a) / log_c(b)

In your case, you want to find log 162:

log 162 = log 2^1 * 3^4

Now, we can use the change of base formula with base 10 (or any other base):

log 162 = (log 2^1 * 3^4) / (log 10)

Since log 10 is simply 1 (logarithm of 10 to any base is 1), we can simplify further:

log 162 = (log 2^1 * 3^4) / 1

Now, apply the properties of logarithms to split the logarithm of a product into a sum of logarithms:

log 162 = (log 2^1) + (log 3^4)

Now, we can replace log 2 with p and log 3 with q:

log 162 = p + (4q)

So, log 162 in terms of p and q is:

log 162 = p + 4q

To write log 162 in terms of p and q, we can use the following steps:

- First, we can write 162 as a product of powers of 2 and 3, such as 162 = 2 x 3^4.

- Next, we can use the property of logarithms that log ab = log a + log b to write log 162 = log 2 + log 3^4.

- Then, we can use another property of logarithms that log a^n = n log a to write log 3^4 = 4 log 3.

- Finally, we can substitute p = log 2 and q = log 3 to get log 162 = p + 4q.