In chemistry, the pH of a substance
can be determined using the formula provided here,
also enlarged here on the right.
Where H plus, is the hydrogen ion concentration
of the substance.
Solutions with a pH less than seven are said to be acidic,
those down here.
And solutions with a pH greater than seven are said
to be basic or alkaline, those up here.
The table below presents the pH values
of some common substances.
I'm going to use the pH function
to answer the following questions.
We first want to determine the hydrogen ion concentration
of tomato juice.
Notice tomato juice has a pH of four.
So to find the hydrogen ion concentration,
we'll substitute four for pH, and solve for H.
So, we'd have four equals negative,
or the opposite of log, H.
And notice how this log does not have a base,
which means it's common log or log base 10.
Now we want to isolate the log with them, then write
the log equation as an exponential equation.
That means we don't want this negative here.
So if it's helpful, we can think of this
as negative one times the common log.
And therefore to isolate the log,
we want to either multiply both sides by negative one,
or divide both sides by negative one.
Which would give us the equation negative four equals log H.
Again, this is common log or log base 10.
And now we'll write this as an exponential equation,
recognizing that the base is 10,
the exponent is negative four,
and the number, or its equal to, H.
So, again, as an exponential equation,
we'd have 10, raise the power of negative four equals H.
So, 10 to the negative four equals
the hydrogen ion concentration.
So we have our answer, but our homework does want
this as a decimal, not as a fraction,
or in exponential form.
So if we have H equals 10 to the negative four,
that's equivalent to one divided by 10 to the fourth.
10 to the fourth equals 10,000.
So we have 1/10,000 as a fraction.
So as a decimal, we'd have zero, point,
zero, zero, zero, one.
We have the tenths, hundredths, thousandths, 10 thousandths.
And notice how our calculator is not much help on this.
If we enter 10 to the negative four, we'd have 10,
exponent negative four.
It gives this scientific notation, not the decimal form.
But we could type in point zero, zero, zero, one,
and see if it's the same in scientific notation, and it is.
So our first answer is zero, point, zero, zero, zero, one.
Next, we'll determine the hydrogen ion concentration
of gastric acid.
Notice that gastric acid has a pH of one.
So to find the hydrogen ion concentration,
we'd have the equation, one equals the opposite
of common log of the hydrogen ion concentration.
Again, the base is 10, so we'll isolate the log with them
by dividing both sides by negative one,
or multiplying both sides by negative one.
Which would give us, negative one equals,
common log, of H.
So again, we have a base, 10, an exponent of negative one,
and the number is the hydrogen ion concentration.
So again, we have 10 to the negative one must equal H.
When bedding a log equation and an exponential equation,
we always start at the base, work our way around
the equal sign, to form the exponential equation.
Again, 10 to the negative one equals H.
So, if the hydrogen ion concentration is equal to 10
to the negative one, that's equivalent to one over 10
to the first, or just one tenth, which as a decimal,
is just zero point one.
So, here's our second answer.
Our last question is different.
If the substance has a hydrogen ion concentration
of one point zero, E, negative nine, which means
one times 10 to the negative ninth, it would have
the same pH as which of these substances.
So now that they're giving us
the hydrogen ion concentration, we want to find the pH.
So our equation would be the pH is equal to the opposite
of the common log of one times 10, to the negative ninth.
Of course, one times 10 to the negative nine,
is just 10 the negative nine.
So we could also write this out as just the opposite
of common log, of 10 to the negative nine.
So what we'll do now, is go to the calculator
and then we'll come back and show how we can evaluate
this without a calculator.
And let's show how to enter both of these.
First we have negative log, and then for our scientific
notation here, one times 10 to the negative ninth,
we enter one, and then second comma for this capital E.
And then we just enter the exponent of negative nine.
Close parentheses and enter.
And notice how the pH is nine.
Or we could also enter negative log, and then 10,
raise the power of negative nine, right arrow,
close parentheses and enter.
Either way the pH is nine.
Now, if we know about the properties of log rhythms,
we could evaluate this without a calculator.
So if the pH is equal to the opposite of common log,
that's heard in this form here of 10 raise
the part of negative nine.
The power property of log rhythms allows us to take
this exponent and move it to the front and multiply it,
by the log rhythm.
So, again, this is the power property of log rhythms.
If you haven't learned this, you can ignore this
for right now.
But applying this property, we'd have the pH,
equals the opposite of negative nine,
and then times the common log of 10.
But the common log of 10 is equal to one.
Notice how the base and the number are the same.
So this does give us a pH of nine.
So regardless of how we approach this,
the pH is nine.
So looking at our table, notice how baking soda
also has a pH of nine.
So whatever this substance is, baking soda has the same pH.
So baking soda is our answer.
Okay, hope you found this helpful.