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If log2=p*log3=q. and log5=r. write an expression for log 150 in terms of p,q, and r.

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Final answer:

To express log150 in terms of p, q, and r, factor 150 into its prime factors (2 × 3 × 5 × 5) and use logarithm properties. The final expression is log 150 = p + q + 2r.

Step-by-step explanation:

To express log150 in terms of p, q, and r, first, we need to factor 150 into its prime factors. The number 150 can be factored into 2 × 3 × 5 × 5. Now we will use the property of logarithms that the logarithm of a product is equal to the sum of the logarithms of the factors (log(ab) = log a + log b) and the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number (log(a^n) = n × log a).

Using these properties, we can write:

log 150 = log(2 × 3 × 5 × 5) = log 2 + log 3 + log(5^2) = log 2 + log 3 + 2 × log 5

Since we know that log2=p, log3=q, and log5=r, we replace the values:

log 150 = p + q + 2r

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